Adjacency Matrix

Adjacency matrixFrom Wikipedia, the free encyclopedia

Contents

1 Adjacency matrix 11.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Adjacency matrix of a bipartite graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Data structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Directed graph 42.1 Basic terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Indegree and outdegree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Degree sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Digraph connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 Classes of digraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Element (mathematics) 103.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Notation and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 Cardinality of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Glossary of graph theory 134.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.1.1 Subgraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.1.2 Walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

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4.1.3 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.1.4 Cliques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.1.5 Strongly connected component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.1.6 Hypercubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.1.7 Knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.1.8 Minors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.1.9 Embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2 Adjacency and degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.2.1 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.3 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.4 Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.5 Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.6 Genus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.7 Weighted graphs and networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.8 Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.8.1 Directed acyclic graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.9 Colouring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.10 Various . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.11 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 Graph (mathematics) 275.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.1.1 Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.1.2 Adjacency relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.2 Types of graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.2.1 Distinction in terms of the main definition . . . . . . . . . . . . . . . . . . . . . . . . . . 285.2.2 Important graph classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.3 Properties of graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.5 Important graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.6 Operations on graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.7 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.11 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.12 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

6 Multigraph 376.1 Undirected multigraph (edges without own identity) . . . . . . . . . . . . . . . . . . . . . . . . . 386.2 Undirected multigraph (edges with own identity) . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

CONTENTS iii

6.3 Directed multigraph (edges without own identity) . . . . . . . . . . . . . . . . . . . . . . . . . . . 386.4 Directed multigraph (edges with own identity) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386.5 Labeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

7 Operations research 417.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

7.2.1 Historical origins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427.2.2 Second World War . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427.2.3 After World War II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

7.3 Problems addressed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447.4 Management science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

7.4.1 Related fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477.4.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

7.5 Societies and journals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

8 Ordered pair 528.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528.2 Defining the ordered pair using set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

8.2.1 Wiener’s definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538.2.2 Hausdorff’s definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538.2.3 Kuratowski definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538.2.4 Quine-Rosser definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558.2.5 Morse definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

8.3 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

9 Orientation (graph theory) 579.1 Oriented graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579.2 Constrained orientations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

10 Path (graph theory) 59

iv CONTENTS

10.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6010.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6010.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6010.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

11 Set (mathematics) 6211.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6311.2 Describing sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6311.3 Membership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

11.3.1 Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6511.3.2 Power sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

11.4 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6611.5 Special sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6611.6 Basic operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

11.6.1 Unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6711.6.2 Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6811.6.3 Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6811.6.4 Cartesian product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

11.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7111.8 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7111.9 Principle of inclusion and exclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7211.10De Morgan’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7211.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7311.12Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7311.13References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7311.14External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

12 Unordered pair 7412.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7412.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7412.3 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 75

12.3.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7512.3.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7712.3.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Chapter 1

Adjacency matrix

In mathematics, computer science and application areas such as sociology, an adjacency matrix is a means of rep-resenting which vertices (or nodes) of a graph are adjacent to which other vertices. Another matrix representationfor a graph is the incidence matrix.Specifically, the adjacency matrix of a finite graph G on n vertices is the n × n matrix where the non-diagonal entryaij is the number of edges from vertex i to vertex j, and the diagonal entry aii, depending on the convention, is eitheronce or twice the number of edges (loops) from vertex i to itself. Undirected graphs often use the latter convention ofcounting loops twice, whereas directed graphs typically use the former convention. There exists a unique adjacencymatrix for each isomorphism class of graphs (up to permuting rows and columns), and it is not the adjacency matrixof any other isomorphism class of graphs. In the special case of a finite simple graph, the adjacency matrix is a(0,1)-matrix with zeros on its diagonal. If the graph is undirected, the adjacency matrix is symmetric.The relationship between a graph and the eigenvalues and eigenvectors of its adjacency matrix is studied in spectralgraph theory.

1.1 Examples

The convention followed here is that an adjacent edge counts 1 in the matrix for an undirected graph.

• The adjacency matrix of a complete graph contains all ones except along the diagonal where there are onlyzeros.

• The adjacency matrix of an empty graph is a zero matrix.

1.2 Adjacency matrix of a bipartite graph

The adjacency matrix A of a bipartite graph whose parts have r and s vertices has the form

A =

(0r,r BBT 0s,s

),

where B is an r× s matrix, and 0 represents the zero matrix. Clearly, the matrix B uniquely represents the bipartitegraphs. It is sometimes called the biadjacency matrix. Formally, let G = (U, V,E) be a bipartite graph with partsU = u1, ..., ur and V = v1, ..., vs . The biadjacency matrix is the r × s 0-1 matrix B in which bi,j = 1 iff(ui, vj) ∈ E .If G is a bipartite multigraph or weighted graph then the elements bi,j are taken to be the number of edges betweenthe vertices or the weight of the edge (ui, vj), respectively.

1

2 CHAPTER 1. ADJACENCY MATRIX

1.3 Properties

The adjacency matrix of an undirected simple graph is symmetric, and therefore has a complete set of real eigenvaluesand an orthogonal eigenvector basis. The set of eigenvalues of a graph is the spectrum of the graph.Suppose two directed or undirected graphsG1 andG2 with adjacency matricesA1 andA2 are given. G1 andG2 areisomorphic if and only if there exists a permutation matrix P such that

PA1P−1 = A2.

In particular, A1 and A2 are similar and therefore have the same minimal polynomial, characteristic polynomial,eigenvalues, determinant and trace. These can therefore serve as isomorphism invariants of graphs. However,two graphs may possess the same set of eigenvalues but not be isomorphic.[1] Such linear operators are said to beisospectral.If A is the adjacency matrix of the directed or undirected graph G, then the matrix An (i.e., the matrix product ofn copies of A) has an interesting interpretation: the entry in row i and column j gives the number of (directed orundirected) walks of length n from vertex i to vertex j. If n is the smallest nonnegative integer, such that for all i ,j, the (i,j)-entry of An > 0, then n is the distance between vertex i and vertex j. This implies, for example, that thenumber of triangles in an undirected graph G is exactly the trace of A3 divided by 6. Note that, the adjacent matrixcan determine whether or not the graph is connected.The main diagonal of every adjacency matrix corresponding to a graph without loops has all zero entries. Note thathere 'loops’ means, for example A→A, not 'cycles’ such as A→B→A.For (d) -regular graphs, d is also an eigenvalue of A for the vector v = (1, . . . , 1) , and G is connected if and onlyif the multiplicity of the eigenvalue d is 1. It can be shown that −d is also an eigenvalue of A if G is a connectedbipartite graph. The above are results of the Perron–Frobenius theorem.

1.4 Variations

An (a, b, c)-adjacency matrix A of a simple graph has Aij = a if ij is an edge, b if it is not, and c on the diagonal.The Seidel adjacency matrix is a (−1,1,0)-adjacency matrix. This matrix is used in studying strongly regular graphsand two-graphs.[2]

The distance matrix has in position (i,j) the distance between vertices vi and vj . The distance is the length ofa shortest path connecting the vertices. Unless lengths of edges are explicitly provided, the length of a path is thenumber of edges in it. The distance matrix resembles a high power of the adjacency matrix, but instead of tellingonly whether or not two vertices are connected (i.e., the connection matrix, which contains boolean values), it givesthe exact distance between them.

1.5 Data structures

For use as a data structure, the main alternative to the adjacency matrix is the adjacency list. Because each entry inthe adjacency matrix requires only one bit, it can be represented in a very compact way, occupying only n2/8 bytesof contiguous space, where n is the number of vertices. Besides avoiding wasted space, this compactness encourageslocality of reference.However, for a sparse graph, adjacency lists require less storage space, because they do not waste any space to rep-resent edges that are not present. Using a naïve array implementation on a 32-bit computer, an adjacency list for anundirected graph requires about 8e bytes of storage, where e is the number of edges.Noting that a simple graph can have at most n2 edges, allowing loops, we can let d = e/n2 denote the density of thegraph. Then, 8e > n2/8 , or the adjacency list representation occupies more space precisely when d > 1/64 . Thusa graph must be sparse indeed to justify an adjacency list representation.Besides the space tradeoff, the different data structures also facilitate different operations. Finding all vertices adjacentto a given vertex in an adjacency list is as simple as reading the list. With an adjacency matrix, an entire row mustinstead be scanned, which takes O(n) time. Whether there is an edge between two given vertices can be determined

1.6. REFERENCES 3

at once with an adjacency matrix, while requiring time proportional to the minimum degree of the two vertices withthe adjacency list.

1.6 References[1] Godsil, Chris; Royle, Gordon Algebraic Graph Theory, Springer (2001), ISBN 0-387-95241-1, p.164

[2] Seidel, J. J. (1968). “Strongly Regular Graphs with (−1,1,0) Adjacency Matrix Having Eigenvalue 3”. Lin. Alg. Appl. 1(2): 281–298. doi:10.1016/0024-3795(68)90008-6.

1.7 Further reading• Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001). “Section 22.1: Repre-sentations of graphs”. Introduction to Algorithms (Second ed.). MIT Press and McGraw-Hill. pp. 527–531.ISBN 0-262-03293-7.

• Godsil, Chris; Royle, Gordon (2001). Algebraic Graph Theory. New York: Springer. ISBN 0-387-95241-1.

1.8 External links• Fluffschack — an educational Java web start game demonstrating the relationship between adjacency matricesand graphs.

• Open Data Structures - Section 12.1 - AdjacencyMatrix: Representing a Graph by a Matrix

• McKay, Brendan. “Description of graph6 and sparse6 encodings”.

• Café math : Adjacency Matrices of Graphs : Application of the adjacency matrices to the computation gen-erating series of walks.

Chapter 2

Directed graph

A directed graph.

In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph, or set of nodesconnected by edges, where the edges have a direction associated with them. In formal terms, a digraph is a pairG = (V,A) (sometimes G = (V,E) ) of:[1]

• a set V, whose elements are called vertices or nodes,

• a set A of ordered pairs of vertices, called arcs, directed edges, or arrows (and sometimes simply edges with

4

2.1. BASIC TERMINOLOGY 5

abcd

1 2 3 41-1 1

1-1

-1-1

1

00 0

000 0 0

a b

cd

1

234

Directed graph with corresponding incidence matrix.

the corresponding set named E instead of A).

It differs from an ordinary or undirected graph, in that the latter is defined in terms of unordered pairs of vertices,which are usually called edges.A digraph is called “simple” if it has no loops, and no multiple arcs (arcs with same starting and ending nodes). Adirected multigraph, in which the arcs constitute a multiset, rather than a set, of ordered pairs of vertices may haveloops (that is, “self-loops” with same starting and ending node) and multiple arcs. Some, but not all, texts allow adigraph, without the qualification simple, to have self loops, multiple arcs, or both.

2.1 Basic terminology

An arc e = (x, y) is considered to be directed from x to y ; y is called the head and x is called the tail of the arc; yis said to be a direct successor of x , and x is said to be a direct predecessor of y . If a path made up of one or moresuccessive arcs leads from x to y , then y is said to be a successor of x , and x is said to be a predecessor of y . Thearc (y, x) is called the arc (x, y) inverted.An orientation of a simple undirected graph is obtained by assigning a direction to each edge. Any directed graphconstructed this way is called an “oriented graph”. A directed graph is an oriented simple graph if and only if it hasneither self-loops nor 2-cycles.[2]

A weighted digraph is a digraph with weights assigned to its arcs, similar to a weighted graph. In the context of graphtheory a digraph with weighted edges is called a network.The adjacency matrix of a digraph (with loops and multiple arcs) is the integer-valued matrix with rows and columnscorresponding to the nodes, where a nondiagonal entry aij is the number of arcs from node i to node j, and thediagonal entry aii is the number of loops at node i. The adjacency matrix of a digraph is unique up to identicalpermutation of rows and columns.Another matrix representation for a digraph is its incidence matrix.See direction for more definitions.

2.2 Indegree and outdegree

For a node, the number of head endpoints adjacent to a node is called the indegree of the node and the number of tailendpoints adjacent to a node is its outdegree (called "branching factor" in trees).

6 CHAPTER 2. DIRECTED GRAPH

A digraph with vertices labeled (indegree, outdegree)

Let G = (V, E) and v ∈ V, then the indegree is denoted deg−(v) and the outdegree as deg+(v). A vertex withdeg−(v) = 0 is called a source, as it is the origin of each of its incident edges. Similarly, a vertex with deg+(v) = 0is called a sink.The degree sum formula states that, for a directed graph,

∑v∈V

deg+(v) =∑v∈V

deg−(v) = |E| .

If, for every node v ∈ V, we have deg+(v) = deg−(v) , the graph is called a balanced digraph.[3]

2.3 Degree sequence

The degree sequence of a directed graph is the list of its indegree and outdegree pairs; for the above example we havedegree sequence ((2,0),(2,2),(0,2),(1,1)). The degree sequence is a directed graph invariant so isomorphic directedgraphs have the same degree sequence. However, the degree sequence does not, in general, uniquely identify a graph;in some cases, non-isomorphic graphs have the same degree sequence.The digraph realization problem is the problem of finding a digraph with the degree sequence being a given sequenceof positive integer pairs. (Trailing pairs of zeros may be ignored since they are trivially realized by adding an appro-priate number of isolated vertices to the digraph.) A sequence which is the degree sequence of some digraph, i.e.for which the digraph realization problem has a solution, is called a digraphic or digraphical sequence. This problemcan either be solved by the Kleitman–Wang algorithm or by the Fulkerson–Chen–Anstee theorem.

2.4 Digraph connectivity

Main article: Connectivity (graph theory)

A digraph G is calledweakly connected (or just connected[4]) if the undirected underlying graph obtained by replacingall directed edges of G with undirected edges is a connected graph. A digraph is strongly connected or strong if it

2.5. CLASSES OF DIGRAPHS 7

contains a directed path from u to v and a directed path from v to u for every pair of vertices u,v. The strong componentsare the maximal strongly connected subgraphs.

2.5 Classes of digraphs

A directed graphG is called symmetric if, for every arc that belongs toG, the corresponding reversed arc also belongsto G. A symmetric, loopless directed graph is equivalent to an undirected graph with the edges replaced by pairs ofinverse arcs; thus the number of edges is equal to the number of arcs halved.

7

11 8

5 3

102 9

A simple acyclic directed graph

An acyclic directed graph, acyclic digraph, or directed acyclic graph is a directed graph with no directed cycles.Special cases of acyclic directed graphs include multitrees (graphs in which no two directed paths from a singlestarting node meet back at the same ending node), oriented trees or polytrees (digraphs formed by orienting the edgesof undirected acyclic graphs), and rooted trees (oriented trees in which all edges of the underlying undirected treeare directed away from the roots).A tournament is an oriented graph obtained by choosing a direction for each edge in an undirected complete graph.In the theory of Lie groups, a quiver Q is a directed graph serving as the domain of, and thus characterizing the shapeof, a representation V defined as a functor, specifically an object of the functor category FinVctKF(Q) where F(Q) isthe free category on Q consisting of paths in Q and FinVctK is the category of finite-dimensional vector spaces overa field K. Representations of a quiver label its vertices with vector spaces and its edges (and hence paths) compatiblywith linear transformations between them, and transform via natural transformations.

8 CHAPTER 2. DIRECTED GRAPH

1 2

3 4A tournament on 4 vertices

2.6 See also• Coates graph

• Flow chart

• Rooted graph

• Flow graph (mathematics)

• Mason graph

• Oriented graph

• Preorder

• Quiver

• Signal-flow graph

• Transpose graph

• Vertical constraint graph

2.7. NOTES 9

2.7 Notes[1] Bang-Jensen & Gutin (2000). Diestel (2005), Section 1.10. Bondy & Murty (1976), Section 10.

[2] Diestel (2005), Section 1.10.

[3] Satyanarayana, Bhavanari; Prasad, Kuncham Syam, Discrete Mathematics and Graph Theory, PHI Learning Pvt. Ltd., p.460, ISBN 978-81-203-3842-5; Brualdi, Richard A. (2006), Combinatorial matrix classes, Encyclopedia of mathematicsand its applications 108, Cambridge University Press, p. 51, ISBN 978-0-521-86565-4.

[4] Bang-Jensen & Gutin (2000) p. 19 in the 2007 edition; p. 20 in the 2nd edition (2009).

2.8 References• Bang-Jensen, Jørgen; Gutin, Gregory (2000), Digraphs: Theory, Algorithms and Applications, Springer, ISBN1-85233-268-9(the corrected 1st edition of 2007 is now freely available on the authors’ site; the 2nd edition appeared in 2009ISBN 1-84800-997-6).

• Bondy, John Adrian; Murty, U. S. R. (1976), Graph Theory with Applications, North-Holland, ISBN 0-444-19451-7.

• Diestel, Reinhard (2005), Graph Theory (3rd ed.), Springer, ISBN 3-540-26182-6 (the electronic 3rd editionis freely available on author’s site).

• Harary, Frank; Norman, Robert Z.; Cartwright, Dorwin (1965), Structural Models: An Introduction to theTheory of Directed Graphs, New York: Wiley.

• Number of directed graphs (or digraphs) with n nodes.

Chapter 3

Element (mathematics)

In mathematics, an element, ormember, of a set is any one of the distinct objects that make up that set.

3.1 Sets

Writing A = {1, 2, 3, 4} means that the elements of the set A are the numbers 1, 2, 3 and 4. Sets of elements of A,for example {1, 2}, are subsets of A.Sets can themselves be elements. For example consider the set B = {1, 2, {3, 4}}. The elements of B are not 1, 2, 3,and 4. Rather, there are only three elements of B, namely the numbers 1 and 2, and the set {3, 4}.The elements of a set can be anything. For example, C = { red, green, blue }, is the set whose elements are the colorsred, green and blue.

3.2 Notation and terminology

First usage of the symbol ϵ in the work Arithmetices principia nova methodo exposita by Giuseppe Peano.

The relation “is an element of”, also called set membership, is denoted by the symbol "∈". Writing

x ∈ A

means that "x is an element of A". Equivalent expressions are "x is a member of A", "x belongs to A", "x is in A"and "x lies in A". The expressions "A includes x" and "A contains x" are also used to mean set membership, howeversome authors use them to mean instead "x is a subset of A".[1] Logician George Boolos strongly urged that “contains”be used for membership only and “includes” for the subset relation only.[2]

Another possible notation for the same relation is

10

3.3. CARDINALITY OF SETS 11

A ∋ x,

meaning "A contains x", though it is used less often.The negation of set membership is denoted by the symbol "∉". Writing

x /∈ A

means that "x is not an element of A".The symbol ϵ was first used by Giuseppe Peano 1889 in his work Arithmetices principia nova methodo exposita. Herehe wrote on page X:

“Signum ϵ significat est. Ita a ϵ b legitur a est quoddam b; ...”

which means

“The symbol ϵ means is. So a ϵ b has to be read as a is a b; ...”

Thereby ϵ is a derivation from the lowercase Greek letter epsilon ("ε") and shall be the first letter of the word ἐστί,which means “is”.The Unicode characters for these symbols are U+2208 ('element of'), U+220B ('contains as member') and U+2209('not an element of'). The equivalent LaTeX commands are "\in”, "\ni” and "\notin”. Mathematica has commands"\[Element]" and "\[NotElement]".

3.3 Cardinality of sets

Main article: Cardinality

The number of elements in a particular set is a property known as cardinality; informally, this is the size of a set. Inthe above examples the cardinality of the set A is 4, while the cardinality of either of the sets B and C is 3. An infiniteset is a set with an infinite number of elements, while a finite set is a set with a finite number of elements. The aboveexamples are examples of finite sets. An example of an infinite set is the set of positive integers = { 1, 2, 3, 4, ... }.

3.4 Examples

Using the sets defined above, namely A = {1, 2, 3, 4 }, B = {1, 2, {3, 4}} and C = { red, green, blue }:

• 2 ∈ A

• {3,4} ∈ B

• {3,4} is a member of B

• Yellow ∉ C

• The cardinality of D = { 2, 4, 8, 10, 12 } is finite and equal to 5.

• The cardinality of P = { 2, 3, 5, 7, 11, 13, ...} (the prime numbers) is infinite (this was proven by Euclid).

3.5 References[1] Eric Schechter (1997). Handbook of Analysis and Its Foundations. Academic Press. ISBN 0-12-622760-8. p. 12

[2] George Boolos (February 4, 1992). 24.243 Classical Set Theory (lecture). (Speech). Massachusetts Institute of Technology,Cambridge, MA.

12 CHAPTER 3. ELEMENT (MATHEMATICS)

3.6 Further reading• Halmos, Paul R. (1974) [1960], Naive Set Theory, Undergraduate Texts in Mathematics (Hardcover ed.), NY:Springer-Verlag, ISBN 0-387-90092-6 - “Naive” means that it is not fully axiomatized, not that it is silly oreasy (Halmos’s treatment is neither).

• Jech, Thomas (2002), “Set Theory”, Stanford Encyclopedia of Philosophy

• Suppes, Patrick (1972) [1960], Axiomatic Set Theory, NY: Dover Publications, Inc., ISBN 0-486-61630-4 -Both the notion of set (a collection of members), membership or element-hood, the axiom of extension, theaxiom of separation, and the union axiom (Suppes calls it the sum axiom) are needed for a more thoroughunderstanding of “set element”.

3.7 External links• Weisstein, Eric W., “Element”, MathWorld.

Chapter 4

Glossary of graph theory

Graph theory is a growing area in mathematical research, and has a large specialized vocabulary. Some authors usethe same word with different meanings. Some authors use different words to mean the same thing. This page attemptsto describe the majority of current usage.

4.1 Basics

A graph G consists of two types of elements, namely vertices and edges. Every edge has two endpoints in the set ofvertices, and is said to connect or join the two endpoints. An edge can thus be defined as a set of two vertices (or anordered pair, in the case of a directed graph - see Section Direction). The two endpoints of an edge are also said tobe adjacent to each other.Alternative models of graphs exist; e.g., a graph may be thought of as a Boolean binary function over the set ofvertices or as a square (0,1)-matrix.A vertex is simply drawn as a node or a dot. The vertex set of G is usually denoted by V(G), or V when there is nodanger of confusion. The order of a graph is the number of its vertices, i.e. |V(G)|.An edge (a set of two elements) is drawn as a line connecting two vertices, called endpoints or end vertices orendvertices. An edge with endvertices x and y is denoted by xy (without any symbol in between). The edge set ofG is usually denoted by E(G), or E when there is no danger of confusion. An edge xy is called incident to a vertexwhen this vertex is one of the endpoints x or y.The size of a graph is the number of its edges, i.e. |E(G)|.[1]

A loop is an edge whose endpoints are the same vertex. A link has two distinct endvertices. An edge is multiple ifthere is another edge with the same endvertices; otherwise it is simple. The multiplicity of an edge is the numberof multiple edges sharing the same end vertices; themultiplicity of a graph, the maximum multiplicity of its edges.A graph is a simple graph if it has no multiple edges or loops, a multigraph if it has multiple edges, but no loops,and amultigraph or pseudograph if it contains both multiple edges and loops (the literature is highly inconsistent).When stated without any qualification, a graph is usually assumed to be simple, except in the literature of categorytheory, where it refers to a quiver.Graphs whose edges or vertices have names or labels are known as labeled, those without as unlabeled. Graphs withlabeled vertices only are vertex-labeled, those with labeled edges only are edge-labeled. The difference between alabeled and an unlabeled graph is that the latter has no specific set of vertices or edges; it is regarded as another wayto look upon an isomorphism type of graphs. (Thus, this usage distinguishes between graphs with identifiable vertexor edge sets on the one hand, and isomorphism types or classes of graphs on the other.)(Graph labeling usually refers to the assignment of labels (usually natural numbers, usually distinct) to the edges andvertices of a graph, subject to certain rules depending on the situation. This should not be confused with a graph’smerely having distinct labels or names on the vertices.)A hyperedge is an edge that is allowed to take on any number of vertices, possibly more than 2. A graph that allowsany hyperedge is called a hypergraph. A simple graph can be considered a special case of the hypergraph, namelythe 2-uniform hypergraph. However, when stated without any qualification, an edge is always assumed to consist of

13

14 CHAPTER 4. GLOSSARY OF GRAPH THEORY

In this pseudograph the blue edges are loops and the red edges are multiple edges of multiplicity 2 and 3. The multiplicity of thegraph is 3.

at most 2 vertices, and a graph is never confused with a hypergraph.A non-edge (or anti-edge) is an edge that is not present in the graph. More formally, for two vertices u and v , {u, v}is a non-edge in a graphG whenever {u, v} is not an edge inG . This means that there is either no edge between thetwo vertices or (for directed graphs) at most one of (u, v) and (v, u) from v is an arc in G.Occasionally the term cotriangle or anti-triangle is used for a set of three vertices none of which are connected.The complement G of a graph G is a graph with the same vertex set as G but with an edge set such that xy is an edgein G if and only if xy is not an edge in G.An edgeless graph or empty graph or null graph is a graph with zero or more vertices, but no edges. The emptygraph or null graph may also be the graph with no vertices and no edges. If it is a graph with no edges and anynumber n of vertices, it may be called the null graph on n vertices. (There is no consistency at all in the literature.)A graph is infinite if it has infinitely many vertices or edges or both; otherwise the graph is finite. An infinite graphwhere every vertex has finite degree is called locally finite. When stated without any qualification, a graph is usuallyassumed to be finite. See also continuous graph.Two graphs G and H are said to be isomorphic, denoted by G ~ H, if there is a one-to-one correspondence, calledan isomorphism, between the vertices of the graph such that two vertices are adjacent in G if and only if their

4.1. BASICS 15

1

23

54

6

A labeled simple graph with vertex set V = {1, 2, 3, 4, 5, 6} and edge set E = {{1,2}, {1,5}, {2,3}, {2,5}, {3,4}, {4,5}, {4,6}}.

corresponding vertices are adjacent in H. Likewise, a graph G is said to be homomorphic to a graph H if thereis a mapping, called a homomorphism, from V(G) to V(H) such that if two vertices are adjacent in G then theircorresponding vertices are adjacent in H.

4.1.1 Subgraphs

A subgraph, H, of a graph, G, is a graph whose vertices are a subset of the vertex set of G, and whose edges are asubset of the edge set of G. In reverse, a supergraph of a graph G is a graph of which G is a subgraph. A graph, G,contains a graph, H, if H is a subgraph of, or is isomorphic to G.A subgraph, H, spans a graph, G, and is a spanning subgraph, or factor of G, if it has the same vertex set as G.A subgraph, H, of a graph, G, is said to be induced (or full) if, for every pair of vertices x and y of H, xy is an edgeof H if and only if xy is an edge of G. In other words, H is an induced subgraph of G if it has exactly the edges thatappear in G over the same vertex set. If the vertex set of H is the subset S of V(G), then H can be written as G[S]and is said to be induced by S.A graph, G, is minimal with some property, P, provided that G has property P and no proper subgraph of G hasproperty P. In this definition, the term subgraph is usually understood to mean induced subgraph. The notion ofmaximality is defined dually: G is maximal with P provided that P(G) and G has no proper supergraph H such thatP(H).A graph that does not contain H as an induced subgraph is said to be H-free, and more generally if F is a family ofgraphs then the graphs that do not contain any induced subgraph isomorphic to a member of F are called F -free.[2]For example the triangle-free graphs are the graphs that do not have a triangle graph as an induced subgraph. Manyimportant classes of graphs can be defined by sets of forbidden subgraphs, the graphs that are not in the class and areminimal with respect to subgraphs, induced subgraphs, or graph minors.A universal graph in a class K of graphs is a simple graph in which every element in K can be embedded as asubgraph.

16 CHAPTER 4. GLOSSARY OF GRAPH THEORY

4.1.2 Walks

A walk is a sequence of vertices and edges, where each edge’s endpoints are the preceding and following vertices inthe sequence. A walk is closed if its first and last vertices are the same, and open if they are different.The length l of a walk is the number of edges that it uses. For an open walk, l = n–1, where n is the number ofvertices visited (a vertex is counted each time it is visited). For a closed walk, l = n (the start/end vertex is listedtwice, but is not counted twice). In the example labeled simple graph, (1, 2, 5, 1, 2, 3) is an open walk with length 5,and (4, 5, 2, 1, 5, 4) is a closed walk of length 5.A trail is a walk in which all the edges are distinct. A closed trail has been called a tour or circuit, but these are notuniversal, and the latter is often reserved for a regular subgraph of degree two.

A directed tour. This is not a simple cycle, since the blue vertices are used twice.

Traditionally, a path referred to what is now usually known as an open walk. Nowadays, when stated without anyqualification, a path is usually understood to be simple, meaning that no vertices (and thus no edges) are repeated.(The term chain has also been used to refer to a walk in which all vertices and edges are distinct.) In the examplelabeled simple graph, (5, 2, 1) is a path of length 2. The closed equivalent to this type of walk, a walk that startsand ends at the same vertex but otherwise has no repeated vertices or edges, is called a cycle. Like path, this termtraditionally referred to any closed walk, but now is usually understood to be simple by definition. In the examplelabeled simple graph, (1, 5, 2, 1) is a cycle of length 3. (A cycle, unlike a path, is not allowed to have length 0.) Pathsand cycles of n vertices are often denoted by Pn and Cn, respectively. (Some authors use the length instead of thenumber of vertices, however.)C1 is a loop, C2 is a digon (a pair of parallel undirected edges in a multigraph, or a pair of antiparallel edges in a

4.1. BASICS 17

directed graph), and C3 is called a triangle.A cycle that has odd length is an odd cycle; otherwise it is an even cycle. One theorem is that a graph is bipartite ifand only if it contains no odd cycles. (See complete bipartite graph.)A graph is acyclic if it contains no cycles; unicyclic if it contains exactly one cycle; and pancyclic if it contains cyclesof every possible length (from 3 to the order of the graph).A wheel graph is a graph with n vertices (n ≥ 4), formed by connecting a single vertex to all vertices of C -₁.The girth of a graph is the length of a shortest (simple) cycle in the graph; and the circumference, the length of alongest (simple) cycle. The girth and circumference of an acyclic graph are defined to be infinity ∞.A path or cycle is Hamiltonian (or spanning) if it uses all vertices exactly once. A graph that contains a Hamil-tonian path is traceable; and one that contains a Hamiltonian path for any given pair of (distinct) end vertices is aHamiltonian connected graph. A graph that contains a Hamiltonian cycle is a Hamiltonian graph.A trail or circuit (or cycle) is Eulerian if it uses all edges precisely once. A graph that contains an Eulerian trail istraversable. A graph that contains an Eulerian circuit is an Eulerian graph.Two paths are internally disjoint (some people call it independent) if they do not have any vertex in common, exceptthe first and last ones.A theta graph is the union of three internally disjoint (simple) paths that have the same two distinct end vertices.[3]A theta0 graph has seven vertices and eight edges that can be drawn as the perimeter and one diameter of a regularhexagon. (The seventh vertex splits the diameter into two edges.) The smallest, excluding multigraphs, topologicalminor of a theta0 graph consists of a square plus one of its diagonals.

4.1.3 Trees

A tree is a connected acyclic simple graph. For directed graphs, each vertex has at most one incoming edge. A vertexof degree 1 is called a leaf, or pendant vertex. An edge incident to a leaf is a leaf edge, or pendant edge. (Somepeople define a leaf edge as a leaf and then define a leaf vertex on top of it. These two sets of definitions are oftenused interchangeably.) A non-leaf vertex is an internal vertex. Sometimes, one vertex of the tree is distinguished,and called the root; in this case, the tree is called rooted. Rooted trees are often treated as directed acyclic graphswith the edges pointing away from the root.A subtree of the tree T is a connected subgraph of T.A forest is an acyclic simple graph. For directed graphs, each vertex has at most one incoming edge. (That is, a treewith the connectivity requirement removed; a graph containing multiple disconnected trees.)A subforest of the forest F is a subgraph of F.A spanning tree is a spanning subgraph that is a tree. Every graph has a spanning forest. But only a connected graphhas a spanning tree.A special kind of tree called a star is K₁,k. An induced star with 3 edges is a claw.A caterpillar is a tree in which all non-leaf nodes form a single path.A k-ary tree is a rooted tree in which every internal vertex has no more than k children. A 1-ary tree is just a path.A 2-ary tree is also called a binary tree.

4.1.4 Cliques

The complete graph Kn of order n is a simple graph with n vertices in which every vertex is adjacent to every other.The pentagon-shaped graph to the right is complete. The complete graph on n vertices is often denoted by Kn. It hasn(n−1)/2 edges (corresponding to all possible choices of pairs of vertices).A clique in a graph is a set of pairwise adjacent vertices. Since any subgraph induced by a clique is a completesubgraph, the two terms and their notations are usually used interchangeably. A k-clique is a clique of order k. In theexample labeled simple graph above, vertices 1, 2 and 5 form a 3-clique, or a triangle. A maximal clique is a cliquethat is not a subset of any other clique (some authors reserve the term clique for maximal cliques).The clique number ω(G) of a graph G is the order of a largest clique in G.

18 CHAPTER 4. GLOSSARY OF GRAPH THEORY

4

1 2

3

5

6A labeled tree with 6 vertices and 5 edges. Nodes 1, 2, 3, and 6 are leaves, while 4 and 5 are internal vertices.

4.1.5 Strongly connected component

A related but weaker concept is that of a strongly connected component. Informally, a strongly connected componentof a directed graph is a subgraph where all nodes in the subgraph are reachable by all other nodes in the subgraph.Reachability between nodes is established by the existence of a path between the nodes.A directed graph can be decomposed into strongly connected components by running the depth-first search (DFS)algorithm twice: first, on the graph itself and next on the transpose graph in decreasing order of the finishing timesof the first DFS. Given a directed graph G, the transpose GT is the graph G with all the edge directions reversed.

4.1. BASICS 19

K5, a complete graph. If a subgraph looks like this, the vertices in that subgraph form a clique of size 5.

4.1.6 Hypercubes

A hypercube graphQn is a regular graph with 2n vertices, 2n−1n edges, and n edges touching each vertex. It can beobtained as the one-dimensional skeleton of the geometric hypercube.

4.1.7 Knots

A knot in a directed graph is a collection of vertices and edges with the property that every vertex in the knot hasoutgoing edges, and all outgoing edges from vertices in the knot terminate at other vertices in the knot. Thus it isimpossible to leave the knot while following the directions of the edges.

4.1.8 Minors

A minor G2 = (V2, E2) of G1 = (V1, E1) is an injection from V2 to V1 such that every edge in E2 corresponds toa path (disjoint from all other such paths) in G1 such that every vertex in V1 is in one or more paths, or is part ofthe injection from V2 to V1 . This can alternatively be phrased in terms of contractions, which are operations whichcollapse a path and all vertices on it into a single edge (see Minor (graph theory)).

20 CHAPTER 4. GLOSSARY OF GRAPH THEORY

4.1.9 Embedding

An embeddingG2 = (V2, E2) ofG1 = (V1, E1) is an injection from V2 to V1 such that every edge inE2 correspondsto a path in G1 .[4]

4.2 Adjacency and degree

In graph theory, degree, especially that of a vertex, is usually a measure of immediate adjacency.An edge connects two vertices; these two vertices are said to be incident to that edge, or, equivalently, that edgeincident to those two vertices. All degree-related concepts have to do with adjacency or incidence.The degree, or valency, dG(v) of a vertex v in a graph G is the number of edges incident to v, with loops beingcounted twice. A vertex of degree 0 is an isolated vertex. A vertex of degree 1 is a leaf. In the example labeledsimple graph, vertices 1 and 3 have a degree of 2, vertices 2, 4 and 5 have a degree of 3, and vertex 6 has a degree of1. If E is finite, then the total sum of vertex degrees is equal to twice the number of edges.The total degree of a graph is the sum of the degrees of all its vertices. Thus, for a graph without loops, it is equal tothe number of incidences between vertices and edges. The handshaking lemma states that the total degree is alwaysequal to two times the number of edges, loops included. This means that for a simple graph with 3 vertices with eachvertex having a degree of two (i.e. a triangle) the total degree would be six (e.g. 3 x 2 = 6).A degree sequence is a list of degrees of a graph in non-increasing order (e.g. d1 ≥ d2 ≥ … ≥ dn). A sequence ofnon-increasing integers is realizable if it is a degree sequence of some graph.Two vertices u and v are called adjacent if an edge exists between them. We denote this by u ~ v or u ↓ v. In theabove graph, vertices 1 and 2 are adjacent, but vertices 2 and 4 are not. The set of neighbors of v, that is, verticesadjacent to v not including v itself, forms an induced subgraph called the (open) neighborhood of v and denotedNG(v). When v is also included, it is called a closed neighborhood and denoted by NG[v]. When stated withoutany qualification, a neighborhood is assumed to be open. The subscript G is usually dropped when there is no dangerof confusion; the same neighborhood notation may also be used to refer to sets of adjacent vertices rather than thecorresponding induced subgraphs. In the example labeled simple graph, vertex 1 has two neighbors: vertices 2 and5. For a simple graph, the number of neighbors that a vertex has coincides with its degree.A dominating set of a graph is a vertex subset whose closed neighborhood includes all vertices of the graph. A vertexv dominates another vertex u if there is an edge from v to u. A vertex subset V dominates another vertex subsetU if every vertex in U is adjacent to some vertex in V. The minimum size of a dominating set is the dominationnumber γ(G).In computers, a finite, directed or undirected graph (with n vertices, say) is often represented by its adjacencymatrix:an n-by-n matrix whose entry in row i and column j gives the number of edges from the i-th to the j-th vertex.Spectral graph theory studies relationships between the properties of a graph and its adjacency matrix or othermatrices associated with the graph.The maximum degree Δ(G) of a graph G is the largest degree over all vertices; the minimum degree δ(G), thesmallest.A graph in which every vertex has the same degree is regular. It is k-regular if every vertex has degree k. A 0-regular graph is an independent set. A 1-regular graph is a matching. A 2-regular graph is a vertex disjoint union ofcycles. A 3-regular graph is said to be cubic, or trivalent.A k-factor is a k-regular spanning subgraph. A 1-factor is a perfect matching. A partition of edges of a graph intok-factors is called a k-factorization. A k-factorable graph is a graph that admits a k-factorization.A graph is biregular if it has unequal maximum andminimum degrees and every vertex has one of those two degrees.A strongly regular graph is a regular graph such that any adjacent vertices have the same number of commonneighbors as other adjacent pairs and that any nonadjacent vertices have the same number of common neighbors asother nonadjacent pairs.

4.3. COMPLEXITY 21

4.2.1 Independence

In graph theory, the word independent usually carries the connotation of pairwise disjoint or mutually nonadjacent.In this sense, independence is a form of immediate nonadjacency. An isolated vertex is a vertex not incident to anyedges. An independent set, or coclique, or stable set, is a set of vertices of which no pair is adjacent. Since the graphinduced by any independent set is an empty graph, the two terms are usually used interchangeably. In the examplelabeled simple graph at the top of this page, vertices 1, 3, and 6 form an independent set; and 2 and 4 form anotherone.Two subgraphs are edge disjoint if they have no edges in common. Similarly, two subgraphs are vertex disjoint ifthey have no vertices (and thus, also no edges) in common. Unless specified otherwise, a set of disjoint subgraphsare assumed to be pairwise vertex disjoint.The independence number α(G) of a graph G is the size of the largest independent set of G.A graph can be decomposed into independent sets in the sense that the entire vertex set of the graph can be partitionedinto pairwise disjoint independent subsets. Such independent subsets are called partite sets, or simply parts.A graph that can be decomposed into two partite sets bipartite; three sets, tripartite; k sets, k-partite; and anunknown number of sets,multipartite. An 1-partite graph is the same as an independent set, or an empty graph. A2-partite graph is the same as a bipartite graph. A graph that can be decomposed into k partite sets is also said to bek-colourable.A complete multipartite graph is a graph in which vertices are adjacent if and only if they belong to different partitesets. A complete bipartite graph is also referred to as a biclique; if its partite sets contain n andm vertices, respectively,then the graph is denoted Kn,m.A k-partite graph is semiregular if each of its partite sets has a uniform degree; equipartite if each partite set hasthe same size; and balanced k-partite if each partite set differs in size by at most 1 with any other.Thematching number α′(G) of a graph G is the size of a largestmatching, or pairwise vertex disjoint edges, of G.A spanning matching, also called a perfect matching is a matching that covers all vertices of a graph.

4.3 Complexity

Complexity of a graph denotes the quantity of information that a graph contained, and can be measured in severalways. For example, by counting the number of its spanning trees, or the value of a certain formula involving thenumber of vertices, edges, and proper paths in a graph. [5]

4.4 Connectivity

Connectivity extends the concept of adjacency and is essentially a form (and measure) of concatenated adjacency.If it is possible to establish a path from any vertex to any other vertex of a graph, the graph is said to be connected;otherwise, the graph is disconnected. A graph is totally disconnected if there is no path connecting any pair ofvertices. This is just another name to describe an empty graph or independent set.A cut vertex, or articulation point, is a vertex whose removal disconnects the remaining subgraph. A cut set, orvertex cut or separating set, is a set of vertices whose removal disconnects the remaining subgraph. A bridge is ananalogous edge (see below).If it is always possible to establish a path from any vertex to every other even after removing any k - 1 vertices,then the graph is said to be k-vertex-connected or k-connected. Note that a graph is k-connected if and only if itcontains k internally disjoint paths between any two vertices. The example labeled simple graph above is connected(and therefore 1-connected), but not 2-connected. The vertex connectivity or connectivity κ(G) of a graph G is theminimum number of vertices that need to be removed to disconnect G. The complete graph Kn has connectivity n -1 for n > 1; and a disconnected graph has connectivity 0.In network theory, a giant component is a connected subgraph that contains a majority of the entire graph’s nodes.A bridge, or cut edge or isthmus, is an edge whose removal disconnects a graph. (For example, all the edges in a treeare bridges.) A cut vertex is an analogous vertex (see above). A disconnecting set is a set of edges whose removal

22 CHAPTER 4. GLOSSARY OF GRAPH THEORY

increases the number of components. An edge cut is the set of all edges which have one vertex in some proper vertexsubset S and the other vertex in V(G)\S. Edges of K3 form a disconnecting set but not an edge cut. Any two edgesof K3 form a minimal disconnecting set as well as an edge cut. An edge cut is necessarily a disconnecting set; and aminimal disconnecting set of a nonempty graph is necessarily an edge cut. A bond is a minimal (but not necessarilyminimum), nonempty set of edges whose removal disconnects a graph.A graph is k-edge-connected if any subgraph formed by removing any k - 1 edges is still connected. The edgeconnectivity κ'(G) of a graph G is the minimum number of edges needed to disconnect G. One well-known result isthat κ(G) ≤ κ'(G) ≤ δ(G).A component is a maximally connected subgraph. A block is either a maximally 2-connected subgraph, a bridge(together with its vertices), or an isolated vertex. A biconnected component is a 2-connected component.An articulation point (also known as a separating vertex) of a graph is a vertex whose removal from the graphincreases its number of connected components. A biconnected component can be defined as a subgraph induced bya maximal set of nodes that has no separating vertex.

4.5 Distance

The distance dG(u, v) between two (not necessary distinct) vertices u and v in a graph G is the length of a shortestpath (also called a graph geodesic) between them. The subscript G is usually dropped when there is no danger ofconfusion. When u and v are identical, their distance is 0. When u and v are unreachable from each other, theirdistance is defined to be infinity ∞.The eccentricity εG(v) of a vertex v in a graph G is the maximum distance from v to any other vertex. The diameterdiam(G) of a graph G is the maximum eccentricity over all vertices in a graph; and the radius rad(G), the minimum.When there are two components in G, diam(G) and rad(G) defined to be infinity ∞. Trivially, diam(G) ≤ 2 rad(G).Vertices with maximum eccentricity are called peripheral vertices. Vertices of minimum eccentricity form thecenter. A tree has at most two center vertices.TheWiener index of a vertex v in a graph G, denoted by WG(v) is the sum of distances between v and all others.TheWiener index of a graph G, denoted byW(G), is the sum of distances over all pairs of vertices. An undirectedgraph’sWiener polynomial is defined to be Σ qd(u,v) over all unordered pairs of vertices u and v. Wiener index andWiener polynomial are of particular interest to mathematical chemists.The k-th power Gk of a graph G is a supergraph formed by adding an edge between all pairs of vertices of G withdistance at most k. A second power of a graph is also called a square.A k-spanner is a spanning subgraph, S, in which every two vertices are at most k times as far apart on S than on G.The number k is the dilation. k-spanner is used for studying geometric network optimization.

4.6 Genus

A crossing is a pair of intersecting edges. A graph is embeddable on a surface if its vertices and edges can bearranged on it without any crossing. The genus of a graph is the lowest genus of any surface on which the graph canembed.A planar graph is one which can be drawn on the (Euclidean) plane without any crossing; and a plane graph, onewhich is drawn in such fashion. In other words, a planar graph is a graph of genus 0. The example labeled simplegraph is planar; the complete graph on n vertices, for n> 4, is not planar. Also, a tree is necessarily a planar graph.When a graph is drawn without any crossing, any cycle that surrounds a region without any edges reaching from thecycle into the region forms a face. Two faces on a plane graph are adjacent if they share a common edge. A dual,or planar dual when the context needs to be clarified, G* of a plane graph G is a graph whose vertices represent thefaces, including any outerface, of G and are adjacent in G* if and only if their corresponding faces are adjacent in G.The dual of a planar graph is always a planar pseudograph (e.g. consider the dual of a triangle). In the familiar caseof a 3-connected simple planar graph G (isomorphic to a convex polyhedron P), the dual G* is also a 3-connectedsimple planar graph (and isomorphic to the dual polyhedron P*).Furthermore, since we can establish a sense of “inside” and “outside” on a plane, we can identify an “outermost”region that contains the entire graph if the graph does not cover the entire plane. Such outermost region is called

4.7. WEIGHTED GRAPHS AND NETWORKS 23

an outer face. An outerplanar graph is one which can be drawn in the planar fashion such that its vertices are alladjacent to the outer face; and an outerplane graph, one which is drawn in such fashion.The minimum number of crossings that must appear when a graph is drawn on a plane is called the crossing number.The minimum number of planar graphs needed to cover a graph is the thickness of the graph.

4.7 Weighted graphs and networks

A weighted graph associates a label (weight) with every edge in the graph. Weights are usually real numbers. Theymay be restricted to rational numbers or integers. Certain algorithms require further restrictions on weights; forinstance, Dijkstra’s algorithm works properly only for positive weights. The weight of a path or the weight of atree in a weighted graph is the sum of the weights of the selected edges. Sometimes a non-edge (a vertex pair withno connecting edge) is indicated by labeling it with a special weight representing infinity. Sometimes the word costis used instead of weight. When stated without any qualification, a graph is always assumed to be unweighted. Insome writing on graph theory the term network is a synonym for a weighted graph. A network may be directed orundirected, it may contain special vertices (nodes), such as source or sink. The classical network problems include:

• minimum cost spanning tree,

• shortest paths,

• maximal flow (and the max-flow min-cut theorem)

4.8 Direction

Main article: Digraph (mathematics)

A directed arc, or directed edge, is an ordered pair of endvertices that can be represented graphically as an arrowdrawn between the endvertices. In such an ordered pair the first vertex is called the initial vertex or tail; the secondone is called the terminal vertex or head (because it appears at the arrow head). An undirected edge disregardsany sense of direction and treats both endvertices interchangeably. A loop in a digraph, however, keeps a sense ofdirection and treats both head and tail identically. A set of arcs aremultiple, or parallel, if they share the same headand the same tail. A pair of arcs are anti-parallel if one’s head/tail is the other’s tail/head. A digraph, or directedgraph, or oriented graph, is analogous to an undirected graph except that it contains only arcs. Amixed graphmaycontain both directed and undirected edges; it generalizes both directed and undirected graphs. When stated withoutany qualification, a graph is almost always assumed to be undirected.A digraph is called simple if it has no loops and at most one arc between any pair of vertices. When stated withoutany qualification, a digraph is usually assumed to be simple. A quiver is a directed graph which is specifically allowed,but not required, to have loops and more than one arc between any pair of vertices.In a digraph Γ, we distinguish the out degree dΓ+(v), the number of edges leaving a vertex v, and the in degreedΓ−(v), the number of edges entering a vertex v. If the graph is oriented, the degree dΓ(v) of a vertex v is equal to thesum of its out- and in- degrees. When the context is clear, the subscript Γ can be dropped. Maximum and minimumout degrees are denoted by Δ+(Γ) and δ+(Γ); and maximum and minimum in degrees, Δ−(Γ) and δ−(Γ).An out-neighborhood, or successor set, N+Γ(v) of a vertex v is the set of heads of arcs going from v. Likewise, anin-neighborhood, or predecessor set, N−Γ(v) of a vertex v is the set of tails of arcs going into v.A source is a vertex with 0 in-degree; and a sink, 0 out-degree.A vertex v dominates another vertex u if there is an arc from v to u. A vertex subset S is out-dominating if everyvertex not in S is dominated by some vertex in S; and in-dominating if every vertex in S is dominated by some vertexnot in S.A kernel in a (possibly directed) graph G is an independent set S such that every vertex in V(G) \ S dominatessome vertex in S. In undirected graphs, kernels are maximal independent sets.[6] A digraph is kernel perfect if everyinduced sub-digraph has a kernel.[7]

An Eulerian digraph is a digraph with equal in- and out-degrees at every vertex.

24 CHAPTER 4. GLOSSARY OF GRAPH THEORY

The zweieck of an undirected edge e = (u, v) is the pair of diedges (u, v) and (v, u)which form the simple dicircuit.An orientation is an assignment of directions to the edges of an undirected or partially directed graph. Whenstated without any qualification, it is usually assumed that all undirected edges are replaced by a directed one in anorientation. Also, the underlying graph is usually assumed to be undirected and simple.A tournament is a digraph in which each pair of vertices is connected by exactly one arc. In other words, it is anoriented complete graph.A directed path, or just a path when the context is clear, is an oriented simple path such that all arcs go the samedirection, meaning all internal vertices have in- and out-degrees 1. A vertex v is reachable from another vertex u ifthere is a directed path that starts from u and ends at v. Note that in general the condition that u is reachable from vdoes not imply that v is also reachable from u.If v is reachable from u, then u is a predecessor of v and v is a successor of u. If there is an arc from u to v, then uis a direct predecessor of v, and v is a direct successor of u.A digraph is strongly connected if every vertex is reachable from every other following the directions of the arcs.On the contrary, a digraph is weakly connected if its underlying undirected graph is connected. A weakly connectedgraph can be thought of as a digraph in which every vertex is “reachable” from every other but not necessarily followingthe directions of the arcs. A strong orientation is an orientation that produces a strongly connected digraph.A directed cycle, or just a cycle when the context is clear, is an oriented simple cycle such that all arcs go the samedirection, meaning all vertices have in- and out-degrees 1. A digraph is acyclic if it does not contain any directedcycle. A finite, acyclic digraph with no isolated vertices necessarily contains at least one source and at least one sink.An arborescence, or out-tree or branching, is an oriented tree in which all vertices are reachable from a single vertex.Likewise, an in-tree is an oriented tree in which a single vertex is reachable from every other one.

4.8.1 Directed acyclic graphs

Main article: directed acyclic graph

The partial order structure of directed acyclic graphs (or DAGs) gives them their own terminology.If there is a directed edge from u to v, then we say u is a parent of v and v is a child of u. If there is a directed pathfrom u to v, we say u is an ancestor of v and v is a descendant of u.The moral graph of a DAG is the undirected graph created by adding an (undirected) edge between all parents ofthe same node (sometimes called marrying), and then replacing all directed edges by undirected edges. A DAG isperfect if, for each node, the set of parents is complete (i.e. no new edges need to be added when forming the moralgraph).

4.9 Colouring

Main article: Graph colouringVertices in graphs can be given colours to identify or label them. Although they may actually be rendered in diagramsin different colours, working mathematicians generally pencil in numbers or letters (usually numbers) to represent thecolours.Given a graph G (V,E) a k-colouring of G is a map ϕ : V → {1, ..., k} with the property that (u, v) ∈ E ⇒ ϕ(u) ≠ϕ(v) - in other words, every vertex is assigned a colour with the condition that adjacent vertices cannot be assignedthe same colour.The chromatic number χ(G) is the smallest k for which G has a k-colouring.Given a graph and a colouring, the colour classes of the graph are the sets of vertices given the same colour.A graph is called k-critical if its chromatic number is k but all of its proper subgraphs have chromatic number lessthan k. An odd cycle is 3-critical, and the complete graph on k vertices is k-critical.

4.10. VARIOUS 25

This graph is an example of a 4-critical graph. Its chromatic number is 4 but all of its proper subgraphs have a chromatic numberless than 4. This graph is also planar

4.10 Various

A graph invariant is a property of a graphG, usually a number or a polynomial, that depends only on the isomorphismclass of G. Examples are the order, genus, chromatic number, and chromatic polynomial of a graph.

4.11 See also

• Graph (mathematics)

• List of graph theory topics

4.12 References[1] Harris, John M. (2000). Combinatorics and Graph Theory. New York: Springer-Verlag. p. 5. ISBN 0-387-98736-3.

[2] Brandstädt, Andreas; Le, Van Bang; Spinrad, Jeremy (1999), “Chapter 7: Forbidden Subgraph”, Graph Classes: A Survey,SIAM Monographs on Discrete Mathematics and Applications, pp. 105–121, ISBN 0-89871-432-X.

[3] Mitchem, John (1969), “Hypo-properties in graphs”, The Many Facets of Graph Theory (Proc. Conf., Western Mich. Univ.,Kalamazoo, Mich., 1968), Springer, pp. 223–230, doi:10.1007/BFb0060121, MR 0253932; Bondy, J. A. (1972), “The“graph theory” of the Greek alphabet”, Graph theory and applications (Proc. Conf., Western Michigan Univ., Kalamazoo,Mich., 1972; dedicated to the memory of J. W. T. Youngs), Lecture Notes in Mathematics 303, Springer, pp. 43–54,doi:10.1007/BFb0067356, MR 0335362.

[4] Rosenberg, Arnold L. and Heath, Lenwood S. (2001). Graph separators with applications. (1st edition ed.). Kluwer. ISBN978-0-306-46464-5.

26 CHAPTER 4. GLOSSARY OF GRAPH THEORY

[5] Neel, David L. (2006), “The linear complexity of a graph”, The electronic journal of combinatorics

[6] Bondy, J.A., Murty, U.S.R., Graph Theory, p. 298

[7] Béla Bollobás, Modern Graph theory, p. 298

• Bollobás, Béla (1998). Modern Graph Theory. Graduate Texts in Mathematics 184. New York: Springer-Verlag. ISBN 0-387-98488-7. Zbl 0902.05016.. [Packed with advanced topics followed by a historicaloverview at the end of each chapter.]

• Brandstädt, Andreas; Le, Van Bang; Spinrad, Jeremy P. (1999). Graph classes: a survey. SIAM Monographson Discrete Mathematics. and Applications 3. Philadelphia, PA: Society for Industrial and Applied Mathe-matics. ISBN 978-0-898714-32-6. Zbl 0919.05001.

• Diestel, Reinhard (2010). Graph Theory. Graduate Texts in Mathematics 173 (4th ed.). Springer-Verlag.ISBN 978-3-642-14278-9. Zbl 1204.05001. [Standard textbook, most basic material and some deeper results,exercises of various difficulty and notes at the end of each chapter; known for being quasi error-free.]

• West, Douglas B. (2001). Introduction to Graph Theory (2ed). Upper Saddle River: Prentice Hall. ISBN0-13-014400-2. [Tons of illustrations, references, and exercises. The most complete introductory guide to thesubject.]

• Weisstein, Eric W., “Graph”, MathWorld.

• Zaslavsky, Thomas. Glossary of signed and gain graphs and allied areas. Electronic Journal of Combinatorics,Dynamic Surveys in Combinatorics, # DS 8. http://www.combinatorics.org/Surveys/

Chapter 5

Graph (mathematics)

This article is about sets of vertices connected by edges. For graphs of mathematical functions, see Graph of afunction. For other uses, see Graph (disambiguation).In mathematics, and more specifically in graph theory, a graph is a representation of a set of objects where some

1

23

54

6

A drawing of a labeled graph on 6 vertices and 7 edges.

pairs of objects are connected by links. The interconnected objects are represented by mathematical abstractionscalled vertices, and the links that connect some pairs of vertices are called edges.[1] Typically, a graph is depicted indiagrammatic form as a set of dots for the vertices, joined by lines or curves for the edges. Graphs are one of theobjects of study in discrete mathematics.The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is anedge between two people if they shake hands, then this is an undirected graph, because if person A shook hands withperson B, then person B also shook hands with person A. In contrast, if there is an edge from person A to person Bwhen person A knows of person B, then this graph is directed, because knowledge of someone is not necessarily asymmetric relation (that is, one person knowing another person does not necessarily imply the reverse; for example,many fans may know of a celebrity, but the celebrity is unlikely to know of all their fans). This latter type of graphis called a directed graph and the edges are called directed edges or arcs.

27

28 CHAPTER 5. GRAPH (MATHEMATICS)

Vertices are also called nodes or points, and edges are also called arcs or lines. Graphs are the basic subject studiedby graph theory. The word “graph” was first used in this sense by J.J. Sylvester in 1878.[2][3]

5.1 Definitions

Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and relatedmathematical structures.

5.1.1 Graph

In the most common sense of the term,[4] a graph is an ordered pair G = (V, E) comprising a set V of vertices ornodes together with a set E of edges or links, which are 2-element subsets of V (i.e., an edge is related with twovertices, and the relation is represented as an unordered pair of the vertices with respect to the particular edge). Toavoid ambiguity, this type of graph may be described precisely as undirected and simple.Other senses of graph stem from different conceptions of the edge set. In one more generalized notion,[5] E is a settogether with a relation of incidence that associates with each edge two vertices. In another generalized notion, E isa multiset of unordered pairs of (not necessarily distinct) vertices. Many authors call this type of object a multigraphor pseudograph.All of these variants and others are described more fully below.The vertices belonging to an edge are called the ends, endpoints, or end vertices of the edge. A vertex may exist ina graph and not belong to an edge.V and E are usually taken to be finite, and many of the well-known results are not true (or are rather different) forinfinite graphs because many of the arguments fail in the infinite case. Moreover, V is often assumed to be non-empty, but E is allowed to be the empty set. The order of a graph is |V | (the number of vertices). A graph’s sizeis |E| , the number of edges. The degree of a vertex is the number of edges that connect to it, where an edge thatconnects to the vertex at both ends (a loop) is counted twice.For an edge {u, v}, graph theorists usually use the somewhat shorter notation uv.

5.1.2 Adjacency relation

The edges E of an undirected graph G induce a symmetric binary relation ~ on V that is called the adjacency relationof G. Specifically, for each edge {u, v} the vertices u and v are said to be adjacent to one another, which is denotedu ~ v.

5.2 Types of graphs

5.2.1 Distinction in terms of the main definition

As stated above, in different contexts it may be useful to refine the term graph with different degrees of generality.Whenever it is necessary to draw a strict distinction, the following terms are used. Most commonly, in modern textsin graph theory, unless stated otherwise, graph means “undirected simple finite graph” (see the definitions below).

A directed graph.

5.2. TYPES OF GRAPHS 29

A simple undirected graph with three vertices and three edges. Each vertex has degree two, so this is also a regulargraph.

Undirected graph

An undirected graph is one in which edges have no orientation. The edge (a, b) is identical to the edge (b, a), i.e., theyare not ordered pairs, but sets {u, v} (or 2-multisets) of vertices. The maximum number of edges in an undirectedgraph without a self-loop is n(n - 1)/2.

Directed graph

Main article: Directed graph

A directed graph or digraph is an ordered pair D = (V, A) with

• V a set whose elements are called vertices or nodes, and

• A a set of ordered pairs of vertices, called arcs, directed edges, or arrows.

An arc a = (x, y) is considered to be directed from x to y; y is called the head and x is called the tail of the arc; y issaid to be a direct successor of x, and x is said to be a direct predecessor of y. If a path leads from x to y, then y issaid to be a successor of x and reachable from x, and x is said to be a predecessor of y. The arc (y, x) is called thearc (x, y) inverted.A directed graph D is called symmetric if, for every arc in D, the corresponding inverted arc also belongs to D. Asymmetric loopless directed graph D = (V, A) is equivalent to a simple undirected graph G = (V, E), where the pairsof inverse arcs in A correspond 1-to-1 with the edges in E; thus the edges in G number |E | = |A|/2, or half the numberof arcs in D.An oriented graph is a directed graph in which at most one of (x, y) and (y, x) may be arcs.

Mixed graph

Main article: Mixed graph

A mixed graph G is a graph in which some edges may be directed and some may be undirected. It is written as anordered triple G = (V, E, A) with V, E, and A defined as above. Directed and undirected graphs are special cases.

Multigraph

A loop is an edge (directed or undirected) which starts and ends on the same vertex; these may be permitted or notpermitted according to the application. In this context, an edge with two different ends is called a link.The term "multigraph" is generally understood to mean that multiple edges (and sometimes loops) are allowed. Wheregraphs are defined so as to allow loops and multiple edges, a multigraph is often defined to mean a graph withoutloops,[6] however, where graphs are defined so as to disallow loops and multiple edges, the term is often defined tomean a “graph” which can have both multiple edges and loops,[7] although many use the term "pseudograph" for thismeaning.[8]

30 CHAPTER 5. GRAPH (MATHEMATICS)

Quiver

A quiver or “multidigraph” is a directed graph which may have more than one arrow from a given source to a giventarget. A quiver may also have directed loops in it.

Simple graph

As opposed to a multigraph, a simple graph is an undirected graph that has no loops (edges connected at both endsto the same vertex) and no more than one edge between any two different vertices. In a simple graph the edges of thegraph form a set (rather than a multiset) and each edge is a pair of distinct vertices. In a simple graph with n vertices,the degree of every vertex is at most n-1.

Weighted graph

A graph is a weighted graph if a number (weight) is assigned to each edge.[9] Such weights might represent, forexample, costs, lengths or capacities, etc. depending on the problem at hand. Some authors call such a graph anetwork.[10] Weighted correlation networks can be defined by soft-thresholding the pairwise correlations amongvariables (e.g. gene measurements).

Half-edges, loose edges

In certain situations it can be helpful to allow edges with only one end, called half-edges, or no ends (loose edges);see for example signed graphs and biased graphs.

5.2.2 Important graph classes

Regular graph

Main article: Regular graph

A regular graph is a graph where each vertex has the same number of neighbours, i.e., every vertex has the samedegree or valency. A regular graph with vertices of degree k is called a k‑regular graph or regular graph of degree k.

Complete graph

Main article: Complete graph

Complete graphs have the feature that each pair of vertices has an edge connecting them.

Finite and infinite graphs

A finite graph is a graph G = (V, E) such that V and E are finite sets. An infinite graph is one with an infinite set ofvertices or edges or both.Most commonly in graph theory it is implied that the graphs discussed are finite. If the graphs are infinite, that isusually specifically stated.

Graph classes in terms of connectivity

Main article: Connectivity (graph theory)

5.3. PROPERTIES OF GRAPHS 31

A complete graph with 5 vertices. Each vertex has an edge to every other vertex.

In an undirected graph G, two vertices u and v are called connected if G contains a path from u to v. Otherwise,they are called disconnected. A graph is called connected if every pair of distinct vertices in the graph is connected;otherwise, it is called disconnected.A graph is called k-vertex-connected or k-edge-connected if no set of k-1 vertices (respectively, edges) exists that,when removed, disconnects the graph. A k-vertex-connected graph is often called simply k-connected.A directed graph is called weakly connected if replacing all of its directed edges with undirected edges producesa connected (undirected) graph. It is strongly connected or strong if it contains a directed path from u to v and adirected path from v to u for every pair of vertices u, v.

Category of all graphs

The category of all graphs is the slice category Set ↓ D where D : Set→ Set is the functor taking a set s to s× s .

5.3 Properties of graphs

See also: Glossary of graph theory and Graph property

32 CHAPTER 5. GRAPH (MATHEMATICS)

Two edges of a graph are called adjacent if they share a common vertex. Two arrows of a directed graph are calledconsecutive if the head of the first one is at the tail of the second one. Similarly, two vertices are called adjacent ifthey share a common edge (consecutive if they are at the tail and at the head of an arrow), in which case the commonedge is said to join the two vertices. An edge and a vertex on that edge are called incident.The graph with only one vertex and no edges is called the trivial graph. A graph with only vertices and no edges isknown as an edgeless graph. The graph with no vertices and no edges is sometimes called the null graph or emptygraph, but the terminology is not consistent and not all mathematicians allow this object.In a weighted graph or digraph, each edge is associated with some value, variously called its cost, weight, length orother term depending on the application; such graphs arise in many contexts, for example in optimal routing problemssuch as the traveling salesman problem.Normally, the vertices of a graph, by their nature as elements of a set, are distinguishable. This kind of graph may becalled vertex-labeled. However, for many questions it is better to treat vertices as indistinguishable; then the graphmay be called unlabeled. (Of course, the vertices may be still distinguishable by the properties of the graph itself,e.g., by the numbers of incident edges). The same remarks apply to edges, so graphs with labeled edges are callededge-labeled graphs. Graphs with labels attached to edges or vertices are more generally designated as labeled.Consequently, graphs in which vertices are indistinguishable and edges are indistinguishable are called unlabeled.(Note that in the literature the term labeled may apply to other kinds of labeling, besides that which serves only todistinguish different vertices or edges.)

5.4 Examples

1

23

54

6

A graph with six nodes.

• The diagram at right is a graphic representation of the following graph:

V = {1, 2, 3, 4, 5, 6}E = {{1, 2}, {1, 5}, {2, 3}, {2, 5}, {3, 4}, {4, 5}, {4, 6}}.

• In category theory a small category can be represented by a directed multigraph in which the objects of the

5.5. IMPORTANT GRAPHS 33

category represented as vertices and the morphisms as directed edges. Then, the functors between categoriesinduce some, but not necessarily all, of the digraph morphisms of the graph.

• In computer science, directed graphs are used to represent knowledge (e.g., Conceptual graph), finite statemachines, and many other discrete structures.

• A binary relation R on a set X defines a directed graph. An element x of X is a direct predecessor of an elementy of X iff xRy.

• A directed edge can model information networks such as Twitter, with one user following another [11]

5.5 Important graphs

Basic examples are:

• In a complete graph, each pair of vertices is joined by an edge; that is, the graph contains all possible edges.

• In a bipartite graph, the vertex set can be partitioned into two sets,W and X, so that no two vertices inW areadjacent and no two vertices in X are adjacent. Alternatively, it is a graph with a chromatic number of 2.

• In a complete bipartite graph, the vertex set is the union of two disjoint sets,W and X, so that every vertex inW is adjacent to every vertex in X but there are no edges withinW or X.

• In a linear graph or path graph of length n, the vertices can be listed in order, v0, v1, ..., v , so that the edgesare vᵢ₋₁vᵢ for each i = 1, 2, ..., n. If a linear graph occurs as a subgraph of another graph, it is a path in thatgraph.

• In a cycle graph of length n ≥ 3, vertices can be named v1, ..., v so that the edges are vᵢ₋₁vi for each i = 2,...,nin addition to v v1. Cycle graphs can be characterized as connected 2-regular graphs. If a cycle graph occursas a subgraph of another graph, it is a cycle or circuit in that graph.

• A planar graph is a graph whose vertices and edges can be drawn in a plane such that no two of the edgesintersect (i.e., embedded in a plane).

• A tree is a connected graph with no cycles.

• A forest is a graph with no cycles (i.e. the disjoint union of one or more trees).

More advanced kinds of graphs are:

• The Petersen graph and its generalizations

• Perfect graphs

• Cographs

• Chordal graphs

• Other graphs with large automorphism groups: vertex-transitive, arc-transitive, and distance-transitive graphs.

• Strongly regular graphs and their generalization distance-regular graphs.

5.6 Operations on graphs

Main article: Operations on graphs

There are several operations that produce new graphs from old ones, which might be classified into the followingcategories:

34 CHAPTER 5. GRAPH (MATHEMATICS)

• Elementary operations, sometimes called “editing operations” on graphs, which create a new graph from theoriginal one by a simple, local change, such as addition or deletion of a vertex or an edge, merging and splittingof vertices, etc.

• Graph rewrite operations replacing the occurrence of some pattern graph within the host graph by an instanceof the corresponding replacement graph.

• Unary operations, which create a significantly new graph from the old one. Examples:

• Line graph• Dual graph• Complement graph

• Binary operations, which create new graph from two initial graphs. Examples:

• Disjoint union of graphs• Cartesian product of graphs• Tensor product of graphs• Strong product of graphs• Lexicographic product of graphs

5.7 Generalizations

In a hypergraph, an edge can join more than two vertices.An undirected graph can be seen as a simplicial complex consisting of 1-simplices (the edges) and 0-simplices (thevertices). As such, complexes are generalizations of graphs since they allow for higher-dimensional simplices.Every graph gives rise to a matroid.In model theory, a graph is just a structure. But in that case, there is no limitation on the number of edges: it can beany cardinal number, see continuous graph.In computational biology, power graph analysis introduces power graphs as an alternative representation of undirectedgraphs.In geographic information systems, geometric networks are closely modeled after graphs, and borrow many conceptsfrom graph theory to perform spatial analysis on road networks or utility grids.

5.8 See also• Conceptual graph

• Dual graph

• Glossary of graph theory

• Graph (data structure)

• Graph database

• Graph drawing

• Graph theory

• Hypergraph

• List of graph theory topics

• List of publications in graph theory

• Network theory

5.9. NOTES 35

5.9 Notes[1] Trudeau, Richard J. (1993). Introduction to Graph Theory (Corrected, enlarged republication. ed.). New York: Dover

Pub. p. 19. ISBN 978-0-486-67870-2. Retrieved 8 August 2012. A graph is an object consisting of two sets called itsvertex set and its edge set.

[2] See:

• J. J. Sylvester (February 7, 1878) “Chemistry and algebra,” Nature, 17 : 284. From page 284: “Every invariant andcovariant thus becomes expressible by a graph precisely identical with a Kekuléan diagram or chemicograph.”

• J. J. Sylvester (1878) “On an application of the new atomic theory to the graphical representation of the invariantsand covariants of binary quantics, — with three appendices,” American Journal of Mathematics, Pure and Applied,1 (1) : 64-90. The term “graph” first appears in this paper on page 65.

[3] Gross, Jonathan L.; Yellen, Jay (2004). Handbook of graph theory. CRC Press. p. 35. ISBN 978-1-58488-090-5.

[4] See, for instance, Iyanaga and Kawada, 69 J, p. 234 or Biggs, p. 4.

[5] See, for instance, Graham et al., p. 5.

[6] For example, see Balakrishnan, p. 1, Gross (2003), p. 4, and Zwillinger, p. 220.

[7] For example, see. Bollobás, p. 7 and Diestel, p. 25.

[8] Gross (1998), p. 3, Gross (2003), p. 205, Harary, p.10, and Zwillinger, p. 220.

[9] Fletcher, Peter; Hoyle, Hughes; Patty, C. Wayne (1991). Foundations of Discrete Mathematics (International student ed.ed.). Boston: PWS-KENT Pub. Co. p. 463. ISBN 0-53492-373-9. A weighted graph is a graph in which a numberw(e), called its weight, is assigned to each edge e.

[10] Strang, Gilbert (2005), Linear Algebra and Its Applications (4th ed.), Brooks Cole, ISBN 0-03-010567-6

[11] Pankaj Gupta, Ashish Goel, Jimmy Lin, Aneesh Sharma, Dong Wang, and Reza Bosagh Zadeh WTF: The who-to-followsystem at Twitter, Proceedings of the 22nd international conference on World Wide Web

5.10 References• Balakrishnan, V. K. (1997-02-01). Graph Theory (1st ed.). McGraw-Hill. ISBN 0-07-005489-4.

• Berge, Claude (1958). Théorie des graphes et ses applications (in French). Dunod, Paris: Collection Universi-taire de Mathématiques, II. pp. viii+277. Translation: -. Dover, New York: Wiley. 2001 [1962].

• Biggs, Norman (1993). Algebraic Graph Theory (2nd ed.). CambridgeUniversity Press. ISBN 0-521-45897-8.

• Bollobás, Béla (2002-08-12). Modern Graph Theory (1st ed.). Springer. ISBN 0-387-98488-7.

• Bang-Jensen, J.; Gutin, G. (2000). Digraphs: Theory, Algorithms and Applications. Springer.

• Diestel, Reinhard (2005). Graph Theory (3rd ed.). Berlin, New York: Springer-Verlag. ISBN 978-3-540-26183-4..

• Graham, R.L., Grötschel, M., and Lovász, L, ed. (1995). Handbook of Combinatorics. MIT Press. ISBN0-262-07169-X.

• Gross, Jonathan L.; Yellen, Jay (1998-12-30). Graph Theory and Its Applications. CRC Press. ISBN 0-8493-3982-0.

• Gross, Jonathan L., & Yellen, Jay, ed. (2003-12-29). Handbook of Graph Theory. CRC. ISBN 1-58488-090-2.

• Harary, Frank (January 1995). Graph Theory. Addison Wesley Publishing Company. ISBN 0-201-41033-8.

• Iyanaga, Shôkichi; Kawada, Yukiyosi (1977). Encyclopedic Dictionary of Mathematics. MIT Press. ISBN0-262-09016-3.

• Zwillinger, Daniel (2002-11-27). CRC Standard Mathematical Tables and Formulae (31st ed.). Chapman &Hall/CRC. ISBN 1-58488-291-3.

36 CHAPTER 5. GRAPH (MATHEMATICS)

5.11 Further reading• Trudeau, Richard J. (1993). Introduction to Graph Theory (Corrected, enlarged republication. ed.). NewYork:Dover Publications. ISBN 978-0-486-67870-2. Retrieved 8 August 2012.

5.12 External links• Weisstein, Eric W., “Graph”, MathWorld.

Chapter 6

Multigraph

This article is about the mathematical concept. For other uses, see Multigraph (disambiguation).“Pseudograph” redirects here. It is not to be confused with Pseudepigraph.In mathematics, and more specifically in graph theory, amultigraph is a graph which is permitted to have multiple

A multigraph with multiple edges (red) and several loops (blue). Not all authors allow multigraphs to have loops.

37

38 CHAPTER 6. MULTIGRAPH

edges (also called parallel edges[1]), that is, edges that have the same end nodes. Thus two vertices may be connectedby more than one edge.There are two distinct notions of multiple edges:

• Edges without own identity: The identity of an edge is defined solely by the two nodes it connects. In this case,the term “multiple edges” means that the same edge can occur several times between these two nodes.

• Edges with own identity: Edges are primitive entities just like nodes. When multiple edges connect two nodes,these are different edges.

A multigraph is different from a hypergraph, which is a graph in which an edge can connect any number of nodes,not just two.For some authors, the terms pseudograph andmultigraph are synonymous. For others, a pseudograph is a multigraphwith loops.

6.1 Undirected multigraph (edges without own identity)

A multigraph G is an ordered pair G:=(V, E) with

• V a set of vertices or nodes,

• E a multiset of unordered pairs of vertices, called edges or lines.

6.2 Undirected multigraph (edges with own identity)

A multigraph G is an ordered triple G:=(V, E, r) with

• V a set of vertices or nodes,

• E a set of edges or lines,

• r : E → {{x,y} : x, y ∈ V}, assigning to each edge an unordered pair of endpoint nodes.

Some authors allow multigraphs to have loops, that is, an edge that connects a vertex to itself,[2] while others callthese pseudographs, reserving the term multigraph for the case with no loops.[3]

6.3 Directed multigraph (edges without own identity)

Amultidigraph is a directed graph which is permitted to havemultiple arcs, i.e., arcs with the same source and targetnodes. A multidigraph G is an ordered pair G:=(V,A) with

• V a set of vertices or nodes,

• A a multiset of ordered pairs of vertices called directed edges, arcs or arrows.

Amixed multigraph G:=(V,E, A) may be defined in the same way as a mixed graph.

6.4 Directed multigraph (edges with own identity)

A multidigraph or quiver G is an ordered 4-tuple G:=(V, A, s, t) with

• V a set of vertices or nodes,

6.5. LABELING 39

• A a set of edges or lines,

• s : A → V , assigning to each edge its source node,

• t : A → V , assigning to each edge its target node.

This notion might be used to model the possible flight connections offered by an airline. In this case the multigraphwould be a directed graph with pairs of directed parallel edges connecting cities to show that it is possible to fly bothto and from these locations.In category theory a small category can be defined as a multidigraph (with edges having their own identity) equippedwith an associative composition law and a distinguished self-loop at each vertex serving as the left and right identityfor composition. For this reason, in category theory the term graph is standardly taken to mean “multidigraph”, andthe underlying multidigraph of a category is called its underlying digraph.

6.5 Labeling

Multigraphs and multidigraphs also support the notion of graph labeling, in a similar way. However there is no unityin terminology in this case.The definitions of labeled multigraphs and labeled multidigraphs are similar, and we define only the latter oneshere.Definition 1: A labeled multidigraph is a labeled graph with labeled arcs.Formally: A labeled multidigraph G is a multigraph with labeled vertices and arcs. Formally it is an 8-tuple G =(ΣV ,ΣA, V, A, s, t, ℓV , ℓA) where

• V is a set of vertices and A is a set of arcs.

• ΣV and ΣA are finite alphabets of the available vertex and arc labels,

• s : A → V and t : A → V are two maps indicating the source and target vertex of an arc,

• ℓV : V → ΣV and ℓA : A → ΣA are two maps describing the labeling of the vertices and arcs.

Definition 2: A labeled multidigraph is a labeled graph with multiple labeled arcs, i.e. arcs with the same end verticesand the same arc label (note that this notion of a labeled graph is different from the notion given by the article graphlabeling).

6.6 See also

• Multidimensional network

• Glossary of graph theory

• Graph theory

6.7 Notes

[1] For example, see Balakrishnan 1997, p. 1 or Chartrand and Zhang 2012, p. 26.

[2] For example, see Bollobás 2002, p. 7 or Diestel 2010, p. 28.

[3] For example, see Wilson 2002, p. 6 or Chartrand and Zhang 2012, pp. 26-27.

40 CHAPTER 6. MULTIGRAPH

6.8 References• Balakrishnan, V. K. (1997). Graph Theory. McGraw-Hill. ISBN 0-07-005489-4.

• Bollobás, Béla (2002). Modern Graph Theory. Graduate Texts in Mathematics 184. Springer. ISBN 0-387-98488-7.

• Chartrand, Gary; Zhang, Ping (2012). A First Course in Graph Theory. Dover. ISBN 978-0-486-48368-9.

• Diestel, Reinhard (2010). Graph Theory. Graduate Texts in Mathematics 173 (4th ed.). Springer. ISBN978-3-642-14278-9.

• Gross, Jonathan L.; Yellen, Jay (1998). Graph Theory and Its Applications. CRC Press. ISBN 0-8493-3982-0.

• Gross, Jonathan L.; Yellen, Jay, eds. (2003). Handbook of Graph Theory. CRC. ISBN 1-58488-090-2.

• Harary, Frank (1995). Graph Theory. Addison Wesley. ISBN 0-201-41033-8.

• Janson, Svante; Knuth, Donald E.; Luczak, Tomasz; Pittel, Boris (1993). “The birth of the giant component”.Random Structures and Algorithms 4 (3): 231–358. doi:10.1002/rsa.3240040303. ISSN 1042-9832. MR1220220.

• Wilson, Robert A. (2002). Graphs, Colourings and the Four-Colour Theorem. Oxford Science Publ. ISBN0-19-851062-4.

• Zwillinger, Daniel (2002). CRC StandardMathematical Tables and Formulae (31st ed.). Chapman&Hall/CRC.ISBN 1-58488-291-3.

6.9 External links• Black, Paul E. “Multigraph”. Dictionary of Algorithms and Data Structures. NIST.

Chapter 7

Operations research

For the academic journal, see Operations Research (journal).

Operations research, or operational research in British usage, is a discipline that deals with the application ofadvanced analytical methods to help make better decisions.[1] It is often considered to be a sub-field of mathematics.[2]The terms management science and decision science are sometimes used as synonyms.[3]

Employing techniques from other mathematical sciences, such as mathematical modeling, statistical analysis, andmathematical optimization, operations research arrives at optimal or near-optimal solutions to complex decision-making problems. Because of its emphasis on human-technology interaction and because of its focus on practicalapplications, operations research has overlap with other disciplines, notably industrial engineering and operationsmanagement, and draws on psychology and organization science. Operations research is often concerned with de-termining the maximum (of profit, performance, or yield) or minimum (of loss, risk, or cost) of some real-worldobjective. Originating in military efforts before World War II, its techniques have grown to concern problems in avariety of industries.[4]

7.1 Overview

Operational research (OR) encompasses a wide range of problem-solving techniques and methods applied in thepursuit of improved decision-making and efficiency, such as simulation, mathematical optimization, queueing theoryand other stochastic-process models, Markov decision processes, econometric methods, data envelopment analysis,neural networks, expert systems, decision analysis, and the analytic hierarchy process.[5] Nearly all of these techniquesinvolve the construction of mathematical models that attempt to describe the system. Because of the computationaland statistical nature of most of these fields, OR also has strong ties to computer science and analytics. Operationalresearchers faced with a new problemmust determine which of these techniques are most appropriate given the natureof the system, the goals for improvement, and constraints on time and computing power.The major subdisciplines in modern operational research, as identified by the journal Operations Research,[6] are:

• Computing and information technologies

•

• Financial engineering

• Manufacturing, service sciences, and supply chain management

• Marketing Engineering[7]

• Policy modeling and public sector work

• Revenue management

• Simulation

41

42 CHAPTER 7. OPERATIONS RESEARCH

• Stochastic models

• Transportation

7.2 History

As a formal discipline, operational research originated in the efforts of military planners during World War II. Inthe decades after the war, the techniques were more widely applied to problems in business, industry and society.Since that time, operational research has expanded into a field widely used in industries ranging from petrochemicalsto airlines, finance, logistics, and government, moving to a focus on the development of mathematical models thatcan be used to analyse and optimize complex systems, and has become an area of active academic and industrialresearch.[4]

7.2.1 Historical origins

Early work in operational research was carried out by individuals such as Charles Babbage. His research into the costof transportation and sorting of mail led to England’s universal “Penny Post” in 1840, and studies into the dynamicalbehaviour of railway vehicles in defence of the GWR's broad gauge.[8] Percy Bridgman brought operational researchto bear on problems in physics in the 1920s and would later attempt to extend these to the social sciences.[9]

Modern operational research originated at the Bawdsey Research Station in the UK in 1937 and was the result of aninitiative of the station’s superintendent, A. P. Rowe. Rowe conceived the idea as a means to analyse and improvethe working of the UK’s early warning radar system, Chain Home (CH). Initially, he analysed the operating of theradar equipment and its communication networks, expanding later to include the operating personnel’s behaviour.This revealed unappreciated limitations of the CH network and allowed remedial action to be taken.[10]

Scientists in the United Kingdom including Patrick Blackett (later Lord Blackett OM PRS), Cecil Gordon, SollyZuckerman, (later Baron Zuckerman OM, KCB, FRS), C. H. Waddington, Owen Wansbrough-Jones, Frank Yates,Jacob Bronowski and Freeman Dyson, and in the United States with George Dantzig looked for ways to make betterdecisions in such areas as logistics and training schedules

7.2.2 Second World War

Themodern field of operational research arose duringWorldWar II. In theWorldWar II era, operational research wasdefined as “a scientific method of providing executive departments with a quantitative basis for decisions regardingthe operations under their control.”[11] Other names for it included operational analysis (UK Ministry of Defencefrom 1962)[12] and quantitative management.[13]

During the SecondWorldWar close to 1,000 men and women in Britain were engaged in operational research. About200 operational research scientists worked for the British Army.[14]

Patrick Blackett worked for several different organizations during the war. Early in the war while working for theRoyal Aircraft Establishment (RAE) he set up a team known as the “Circus” which helped to reduce the number ofanti-aircraft artillery rounds needed to shoot down an enemy aircraft from an average of over 20,000 at the start ofthe Battle of Britain to 4,000 in 1941.[15]

In 1941 Blackett moved from the RAE to the Navy, after first working with RAFCoastal Command, in 1941 and thenearly in 1942 to the Admiralty.[16] Blackett’s team at Coastal Command’s Operational Research Section (CC-ORS)included two future Nobel prize winners and many other people who went on to be pre-eminent in their fields.[17]They undertook a number of crucial analyses that aided the war effort. Britain introduced the convoy system to reduceshipping losses, but while the principle of using warships to accompany merchant ships was generally accepted, it wasunclear whether it was better for convoys to be small or large. Convoys travel at the speed of the slowest member,so small convoys can travel faster. It was also argued that small convoys would be harder for German U-boats todetect. On the other hand, large convoys could deploy more warships against an attacker. Blackett’s staff showed thatthe losses suffered by convoys depended largely on the number of escort vessels present, rather than the size of theconvoy. Their conclusion was that a few large convoys are more defensible than many small ones.[18]

While performing an analysis of the methods used by RAF Coastal Command to hunt and destroy submarines, oneof the analysts asked what colour the aircraft were. As most of them were from Bomber Command they were painted

7.2. HISTORY 43

A Liberator in standard RAF green/dark earth/black night bomber finish as originally used by Coastal Command

black for night-time operations. At the suggestion of CC-ORS a test was run to see if that was the best colour tocamouflage the aircraft for daytime operations in the grey North Atlantic skies. Tests showed that aircraft paintedwhite were on average not spotted until they were 20% closer than those painted black. This change indicated that30% more submarines would be attacked and sunk for the same number of sightings.[19] As a result of these findingsCoastal Command changed their aircraft to using white undersurfaces.

A Warwick in the revised RAF Coastal Command green/dark grey/white finish

44 CHAPTER 7. OPERATIONS RESEARCH

Other work by the CC-ORS indicated that on average if the trigger depth of aerial-delivered depth charges (DCs)were changed from 100 feet to 25 feet, the kill ratios would go up. The reason was that if a U-boat saw an aircraftonly shortly before it arrived over the target then at 100 feet the charges would do no damage (because the U-boatwouldn't have had time to descend as far as 100 feet), and if it saw the aircraft a long way from the target it had timeto alter course under water so the chances of it being within the 20-foot kill zone of the charges was small. It wasmore efficient to attack those submarines close to the surface when the targets’ locations were better known than toattempt their destruction at greater depths when their positions could only be guessed. Before the change of settingsfrom 100 feet to 25 feet, 1% of submerged U-boats were sunk and 14% damaged. After the change, 7% were sunkand 11% damaged. (If submarines were caught on the surface, even if attacked shortly after submerging, the numbersrose to 11% sunk and 15% damaged). Blackett observed “there can be few cases where such a great operational gainhad been obtained by such a small and simple change of tactics”.[20]

Bomber Command’s Operational Research Section (BC-ORS), analysed a report of a survey carried out by RAFBomber Command. For the survey, Bomber Command inspected all bombers returning from bombing raids overGermany over a particular period. All damage inflicted by German air defences was noted and the recommendationwas given that armour be added in the most heavily damaged areas. This recommendation was not adopted becausethe fact that the aircraft returned with these areas damaged indicated these areas were NOT vital, and adding armourto non-vital areas where damage is acceptable negatively affects aircraft performance. Their suggestion to removesome of the crew so that an aircraft loss would result in fewer personnel losses, was also rejected by RAF command.Blackett’s team made the logical recommendation that the armour be placed in the areas which were completelyuntouched by damage in the bombers which returned. They reasoned that the survey was biased, since it only includedaircraft that returned to Britain. The untouched areas of returning aircraft were probably vital areas, which, if hit,would result in the loss of the aircraft.[21]

When Germany organised its air defences into the Kammhuber Line, it was realised by the British that if the RAFbombers were to fly in a bomber stream they could overwhelm the night fighters who flew in individual cells directedto their targets by ground controllers. It was then a matter of calculating the statistical loss from collisions against thestatistical loss from night fighters to calculate how close the bombers should fly to minimise RAF losses.[22]

The “exchange rate” ratio of output to input was a characteristic feature of operational research. By comparing thenumber of flying hours put in by Allied aircraft to the number of U-boat sightings in a given area, it was possible toredistribute aircraft to more productive patrol areas. Comparison of exchange rates established “effectiveness ratios”useful in planning. The ratio of 60 mines laid per ship sunk was common to several campaigns: German mines inBritish ports, British mines on German routes, and United States mines in Japanese routes.[23]

Operational research doubled the on-target bomb rate of B-29s bombing Japan from the Marianas Islands by increas-ing the training ratio from 4 to 10 percent of flying hours; revealed that wolf-packs of three United States submarineswere the most effective number to enable all members of the pack to engage targets discovered on their individualpatrol stations; revealed that glossy enamel paint was more effective camouflage for night fighters than traditional dullcamouflage paint finish, and the smooth paint finish increased airspeed by reducing skin friction.[23]

On land, the operational research sections of the Army Operational Research Group (AORG) of the Ministry ofSupply (MoS) were landed in Normandy in 1944, and they followed British forces in the advance across Europe.They analysed, among other topics, the effectiveness of artillery, aerial bombing and anti-tank shooting.

7.2.3 After World War II

With expanded techniques and growing awareness of the field at the close of the war, operational research was nolonger limited to only operational, but was extended to encompass equipment procurement, training, logistics andinfrastructure. Operations Research also grew in many areas other than the military once scientists learned to applyits principles to the civilian sector. With the development of the simplex algorithm for Linear Programming in 1947[24] and the development of computers over the next three decades, Operations Research can now “solve problemswith hundreds of thousands of variables and constraints. Moreover, the large volumes of data required for suchproblems can be stored and manipulated very efficiently.” [24]

7.3 Problems addressed

• Critical path analysis or project planning: identifying those processes in a complex project which affect theoverall duration of the project

7.3. PROBLEMS ADDRESSED 45

Map of Kammhuber Line

• Floorplanning: designing the layout of equipment in a factory or components on a computer chip to reducemanufacturing time (therefore reducing cost)

• Network optimization: for instance, setup of telecommunications networks to maintain quality of service dur-ing outages

46 CHAPTER 7. OPERATIONS RESEARCH

• Allocation problems

• Facility location

• Assignment Problems:

• Assignment problem• Generalized assignment problem• Quadratic assignment problem• Weapon target assignment problem

• Bayesian search theory : looking for a target

• Optimal search

• Routing, such as determining the routes of buses so that as few buses are needed as possible

• Supply chain management: managing the flow of raw materials and products based on uncertain demand forthe finished products

• Efficient messaging and customer response tactics

• Automation: automating or integrating robotic systems in human-driven operations processes

• Globalization: globalizing operations processes in order to take advantage of cheaper materials, labor, land orother productivity inputs

• Transportation: managing freight transportation and delivery systems (Examples: LTL Shipping, intermodalfreight transport, travelling salesman problem)

• Scheduling:

• Personnel staffing• Manufacturing steps• Project tasks• Network data traffic: these are known as queueing models or queueing systems.• Sports events and their television coverage

• Blending of raw materials in oil refineries

• Determining optimal prices, in many retail and B2B settings, within the disciplines of pricing science

Operational research is also used extensively in government where evidence-based policy is used.

7.4 Management science

Main article: Management science

In 1967 Stafford Beer characterized the field of management science as “the business use of operations research”.[25]However, inmodern times the termmanagement sciencemay also be used to refer to the separate fields of organizationalstudies or corporate strategy. Like operational research itself, management science (MS) is an interdisciplinarybranch of applied mathematics devoted to optimal decision planning, with strong links with economics, business, en-gineering, and other sciences. It uses various scientific research-based principles, strategies, and analytical methodsincluding mathematical modeling, statistics and numerical algorithms to improve an organization’s ability to enactrational and meaningful management decisions by arriving at optimal or near optimal solutions to complex deci-sion problems. In short, management sciences help businesses to achieve their goals using the scientific methods ofoperational research.

7.5. SOCIETIES AND JOURNALS 47

The management scientist’s mandate is to use rational, systematic, science-based techniques to inform and improvedecisions of all kinds. Of course, the techniques of management science are not restricted to business applications butmay be applied to military, medical, public administration, charitable groups, political groups or community groups.Management science is concerned with developing and applying models and concepts that may prove useful in helpingto illuminate management issues and solve managerial problems, as well as designing and developing new and bettermodels of organizational excellence.[26]

The application of these models within the corporate sector became known as management science.[27]

7.4.1 Related fields

Some of the fields that have considerable overlap with Operations Research and Management Science include:

7.4.2 Applications

Applications of management science is abundant in industry as airlines, manufacturing companies, service organi-zations, military branches, and in government. The range of problems and issues to which management science hascontributed insights and solutions is vast. It includes:[26]

• scheduling airlines, including both planes and crew,• deciding the appropriate place to site new facilities such as a warehouse, factory or fire station,• managing the flow of water from reservoirs,• identifying possible future development paths for parts of the telecommunications industry,• establishing the information needs and appropriate systems to supply them within the health service, and• identifying and understanding the strategies adopted by companies for their information systems

Management science is also concerned with so-called ”soft-operational analysis”, which concerns methods for strate-gic planning, strategic decision support, and Problem Structuring Methods (PSM). In dealing with these sorts ofchallenges mathematical modeling and simulation are not appropriate or will not suffice. Therefore, during the past30 years, a number of non-quantified modeling methods have been developed. These include:

• stakeholder based approaches including metagame analysis and drama theory• morphological analysis and various forms of influence diagrams.• approaches using cognitive mapping• the Strategic Choice Approach• robustness analysis

7.5 Societies and journalsSocieties

The International Federation of Operational Research Societies (IFORS)[28] is an umbrella organization for opera-tional research societies worldwide, representing approximately 50 national societies including those in the US,[29]UK,[30] France,[31] Germany, Canada,[32] Australia,[33] New Zealand,[34] Philippines,[35] India,[36] Japan and SouthAfrica (ORSSA).[37] The constituent members of IFORS form regional groups, such as that in Europe.[38] Otherimportant operational research organizations are Simulation Interoperability Standards Organization (SISO)[39] andInterservice/Industry Training, Simulation and Education Conference (I/ITSEC)[40]

In 2004 the US-based organization INFORMS began an initiative to market the OR profession better, including awebsite entitledThe Science of Better[41] which provides an introduction to OR and examples of successful applicationsof OR to industrial problems. This initiative has been adopted by the Operational Research Society in the UK,including a website entitled Learn about OR.[42]

48 CHAPTER 7. OPERATIONS RESEARCH

Journals

The Institute for Operations Research and the Management Sciences (INFORMS) publishes thirteen scholarly jour-nals about operations research, including the top two journals in their class, according to 2005 Journal CitationReports.[43] They are:

• Decision Analysis

• Information Systems Research

• INFORMS Journal on Computing

• INFORMS Transactions on Education[44] (an open access journal)

• Interfaces: An International Journal of the Institute for Operations Research and the Management Sciences

• Management Science: A Journal of the Institute for Operations Research and the Management Sciences

• Manufacturing & Service Operations Management

• Marketing Science

• Mathematics of Operations Research

• Operations Research: A Journal of the Institute for Operations Research and the Management Sciences

• Organization Science

• Service Science

• Transportation Science.

Other journals

• 4OR-A Quarterly Journal of Operations Research: jointly published the Belgian, French and Italian OperationsResearch Societies (Springer);

• Decision Sciences published by Wiley-Blackwell on behalf of the Decision Sciences Institute

• European Journal of Operational Research (EJOR): Founded in 1975 and is presently by far the largest opera-tional research journal in the world, with its around 9,000 pages of published papers per year. In 2004, its totalnumber of citations was the second largest amongst Operational Research and Management Science journals;

• INFOR Journal: published and sponsored by the Canadian Operational Research Society;

• International Journal of Operations Research and Information Systems (IJORIS)": an official publication of theInformation Resources Management Association, published quarterly by IGI Global;[45]

• Journal of Defense Modeling and Simulation (JDMS): Applications, Methodology, Technology: a quarterly jour-nal devoted to advancing the science of modeling and simulation as it relates to the military and defense.[46]

• Journal of the Operational Research Society (JORS): an official journal of The OR Society; this is the oldestcontinuously published journal of OR in the world, published by Palgrave;[47]

• Journal of Simulation (JOS): an official journal of The OR Society, published by Palgrave;[47]

• Mathematical Methods of Operations Research (MMOR): the journal of the German and Dutch OR Societies,published by Springer;[48]

• Military Operations Research (MOR): published by the Military Operations Research Society;

• Opsearch: official journal of the Operational Research Society of India;

• OR Insight: a quarterly journal of The OR Society, published by Palgrave;[47]

• Production and Operations Management, the official journal of the Production and Operations ManagementSociety

• TOP: the official journal of the Spanish Society of Statistics and Operations Research.[49]

7.6. SEE ALSO 49

7.6 See also

7.7 References[1] “About Operations Research”. INFORMS.org. Retrieved 7 January 2012.

[2] “Mathematics Subject Classification”. American Mathematical Society. 23 May 2011. Retrieved 7 January 2012.

[3] Wetherbe, James C. (1979), Systems analysis for computer-based information systems, West series in data processing andinformation systems, West Pub. Co., ISBN 9780829902280, A systems analyst who contributes in the area of DSS mustbe skilled in such areas as management science (synonymous with decision science and operations research), modeling,simulation, and advanced statistics.

[4] “What is OR”. HSOR.org. Retrieved 13 November 2011.

[5] “Operations Research Analysts”. Bls.gov. Retrieved 27 January 2012.

[6] “OR / Pubs / IOL Home”. INFORMS.org. 2 January 2009. Retrieved 13 November 2011.

[7] “DecisionPro, Inc. – Makers of Marketing Engineering Software – Home Page”. Decisionpro.biz. Retrieved 13 November2011.

[8] M.S. Sodhi, “What about the 'O' in O.R.?" OR/MS Today, December, 2007, p. 12, http://www.lionhrtpub.com/orms/orms-12-07/frqed.html

[9] P. W. Bridgman, The Logic of Modern Physics, The MacMillan Company, New York, 1927

[10] “operations research (industrial engineering) :: History – Britannica Online Encyclopedia”. Britannica.com. Retrieved 13November 2011.

[11] “Operational Research in the British Army 1939–1945, October 1947, Report C67/3/4/48, UK National Archives fileWO291/1301Quoted on the dust-jacket of: Morse, Philip M, and Kimball, George E, Methods of Operations Research, 1st EditionRevised, pub MIT Press & J Wiley, 5th printing, 1954.

[12] UK National Archives Catalogue for WO291 lists a War Office organisation called Army Operational Research Group(AORG) that existed from 1946 to 1962. “In January 1962 the name was changed to Army Operational Research Estab-lishment (AORE). Following the creation of a unified Ministry of Defence, a tri-service operational research organisationwas established: the Defence Operational Research Establishment (DOAE) which was formed in 1965, and it the ArmyOperational Research Establishment based at West Byfleet.”

[13] http://brochure.unisa.ac.za/myunisa/data/subjects/Quantitative%20Management.pdf

[14] Kirby, p. 117

[15] Kirby, pp. 91–94

[16] Kirby, p. 96,109

[17] Kirby, p. 96

[18] ""Numbers are Essential": Victory in the North Atlantic Reconsidered, March–May 1943”. Familyheritage.ca. 24 May1943. Retrieved 13 November 2011.

[19] Kirby, p. 101

[20] (Kirby, pp. 102,103)

[21] James F. Dunnigan (1999). Dirty Little Secrets of the Twentieth Century. Harper Paperbacks. pp. 215–217.

[22] “RAF History – Bomber Command 60th Anniversary”. Raf.mod.uk. Retrieved 13 November 2011.

[23] Milkman, Raymond H. (May 1968). “Operations Research in World War II”. United States Naval Institute Proceedings.

[24] http://www.pitt.edu/~{}jrclass/or/or-intro.html#history

[25] Stafford Beer (1967) Management Science: The Business Use of Operations Research

[26] What is Management Science? Lancaster University, 2008. Retrieved 5 June 2008.

50 CHAPTER 7. OPERATIONS RESEARCH

[27] What is Management Science? The University of Tennessee, 2006. Retrieved 5 June 2008.

[28] “IFORS”. IFORS. Retrieved 13 November 2011.

[29] Leszczynski, Mary (8 November 2011). “Informs”. Informs. Retrieved 13 November 2011.

[30] “The OR Society”. Orsoc.org.uk. Retrieved 13 November 2011.

[31] “Société française de Recherche Opérationnelle et d'Aide à la Décision”. ROADEF. Retrieved 13 November 2011.

[32] www.cors.ca. “CORS”. Cors.ca. Retrieved 13 November 2011.

[33] “ASOR”. ASOR. 1 January 1972. Retrieved 13 November 2011.

[34] “ORSNZ”. ORSNZ. Retrieved 13 November 2011.

[35] “ORSP”. ORSP. Retrieved 13 November 2011.

[36] “ORSI”. Orsi.in. Retrieved 13 November 2011.

[37] “ORSSA”. ORSSA. 23 September 2011. Retrieved 13 November 2011.

[38] “EURO”. Euro-online.org. Retrieved 13 November 2011.

[39] “SISO”. Sisostds.org. Retrieved 13 November 2011.

[40] “I/Itsec”. I/Itsec. Retrieved 13 November 2011.

[41] “The Science of Better”. The Science of Better. Retrieved 13 November 2011.

[42] “Learn about OR”. Learn about OR. Retrieved 13 November 2011.

[43] “INFORMS Journals”. Informs.org. Retrieved 13 November 2011.

[44] “INFORMS Transactions on Education”. Informs.org. Retrieved 19 March 2015.

[45] “International Journal of Operations Research and Information Systems (IJORIS) (1947–9328)(1947–9336): John Wang:Journals”. IGI Global. Retrieved 13 November 2011.

[46] The Society for Modeling & Simulation International. “JDMS”. Scs.org. Retrieved 13 November 2011.

[47] The OR Society;

[48] “Mathematical Methods of Operations Research website”. Springer.com. Retrieved 13 November 2011.

[49] “TOP”. Springer.com. Retrieved 13 November 2011.

7.8 Notes• Kirby, M. W. (Operational Research Society (Great Britain)). Operational Research in War and Peace: TheBritish Experience from the 1930s to 1970, Imperial College Press, 2003. ISBN 1-86094-366-7, ISBN 978-1-86094-366-9

7.9 Further reading• C. West Churchman, Russell L. Ackoff & E. L. Arnoff, Introduction to Operations Research, New York: J.Wiley and Sons, 1957

• Joseph G. Ecker & Michael Kupferschmid, Introduction to Operations Research, Krieger Publishing Co.

• Frederick S. Hillier & Gerald J. Lieberman, Introduction to Operations Research, McGraw-Hill: Boston MA;8th. (International) Edition, 2005

• Michael Pidd, Tools for Thinking: Modelling in Management Science, J. Wiley & Sons Ltd., Chichester; 2nd.Edition, 2003

• Hamdy A. Taha, Operations Research: An Introduction, Prentice Hall; 9th. Edition, 2011

7.10. EXTERNAL LINKS 51

• Wayne Winston, Operations Research: Applications and Algorithms, Duxbury Press; 4th. Edition, 2003

• Kenneth R. Baker, Dean H. Kropp (1985). Management Science: An Introduction to the Use of Decision Models

• David Charles Heinze (1982). Management Science: Introductory Concepts and Applications

• Lee J. Krajewski, Howard E. Thompson (1981). “Management Science: Quantitative Methods in Context”

• Thomas W. Knowles (1989). Management science: Building and Using Models

• Kamlesh Mathur, Daniel Solow (1994). Management Science: The Art of Decision Making

• Laurence J. Moore, Sang M. Lee, Bernard W. Taylor (1993). Management Science

• William Thomas Morris (1968). Management Science: A Bayesian Introduction.

• William E. Pinney, Donald B. McWilliams (1987). Management Science: An Introduction to QuantitativeAnalysis for Management

• Shuchman, Abraham. Scientific decision making in business. New York: Holt, Rinehart and Winston (1963)

• Shrader, Charles R. (2006). History of Operations Research in the United States Army, Volume 1:1942–1962.Washington, D.C.: United States Army Center of Military History. CMH Pub 70-102-1.

• Gerald E. Thompson (1982). Management Science: An Introduction to Modern Quantitative Analysis and De-cision Making. New York : McGraw-Hill Publishing Co.

• Saul I. Gass & Arjang A. Assad (2005). An Annotated Timeline of Operations Research: An Informal History.New York : Kluwer Academic Publishers.

• C. H. Waddington. “O. R. in World War 2: Operational Research against the U-boat”, Elek Science, London,1973.

7.10 External links• What is Operations Research?

• International Federation of Operational Research Societies

• The Institute for Operations Research and the Management Sciences (INFORMS)

• Occupational Outlook Handbook, U.S. Department of Labor Bureau of Labor Statistics

• “Operation Everything: It Stocks Your Grocery Store, Schedules Your Favorite Team’s Games, and Helps PlanYour Vacation. The Most Influential Academic Discipline You've Never Heard Of.” Boston Globe, 27 June2004

• “Optimal Results: IT-powered advances in operations research can enhance business processes and boost thecorporate bottom line.” Computerworld, 20 November 2000

Chapter 8

Ordered pair

In mathematics, an ordered pair (a, b) is a pair of mathematical objects. The order in which the objects appear inthe pair is significant: the ordered pair (a, b) is different from the ordered pair (b, a) unless a = b. (In contrast, theunordered pair {a, b} equals the unordered pair {b, a}.)Ordered pairs are also called 2-tuples, or sequences of length 2; ordered pairs of scalars are also called 2-dimensionalvectors. The entries of an ordered pair can be other ordered pairs, enabling the recursive definition of ordered n-tuples(ordered lists of n objects). For example, the ordered triple (a,b,c) can be defined as (a, (b,c)), i.e., as one pair nestedin another.In the ordered pair (a, b), the object a is called the first entry, and the object b the second entry of the pair. Al-ternatively, the objects are called the first and second coordinates, or the left and right projections of the orderedpair.Cartesian products and binary relations (and hence functions) are defined in terms of ordered pairs.

8.1 Generalities

Let (a1, b1) and (a2, b2) be ordered pairs. Then the characteristic (or defining) property of the ordered pair is:

(a1, b1) = (a2, b2) if only and if a1 = a2 and b1 = b2.

The set of all ordered pairs whose first entry is in some set A and whose second entry is in some set B is called theCartesian product of A and B, and written A × B. A binary relation between sets A and B is a subset of A × B.If one wishes to employ the (a, b) notation for a different purpose (such as denoting open intervals on the real numberline) the ordered pair may be denoted by the variant notation ⟨a, b⟩ .The left and right projection of a pair p is usually denoted by π1(p) and π2(p), or by πl(p) and πr(p), respectively. Incontexts where arbitrary n-tuples are considered, πni(t) is a common notation for the i-th component of an n tuple t.

8.2 Defining the ordered pair using set theory

The above characteristic property of ordered pairs is all that is required to understand the role of ordered pairs inmathematics. Hence the ordered pair can be taken as a primitive notion, whose associated axiom is the characteristicproperty. This was the approach taken by the N. Bourbaki group in its Theory of Sets, published in 1954, long afterKuratowski discovered his reduction (below). The Kuratowski definition was added in the second edition of Theoryof Sets, published in 1970.If one agrees that set theory is an appealing foundation of mathematics, then all mathematical objects must be definedas sets of some sort. Hence if the ordered pair is not taken as primitive, it must be defined as a set.[1] Several set-theoretic definitions of the ordered pair are given below.

52

8.2. DEFINING THE ORDERED PAIR USING SET THEORY 53

8.2.1 Wiener’s definition

Norbert Wiener proposed the first set theoretical definition of the ordered pair in 1914:[2]

(a, b) := {{{a} , ∅} , {{b}}} .

He observed that this definition made it possible to define the types of Principia Mathematica as sets. PrincipiaMathematica had taken types, and hence relations of all arities, as primitive.Wiener used {{b}} instead of {b} to make the definition compatible with type theory where all elements in a classmust be of the same “type”. With nesting b within an additional set its type is made equal to {{a}, ∅} 's.

8.2.2 Hausdorff’s definition

About the same time as Wiener (1914), Felix Hausdorff proposed his definition:

(a, b) := {{a, 1}, {b, 2}}

“where 1 and 2 are two distinct objects different from a and b.”[3]

8.2.3 Kuratowski definition

In 1921 Kazimierz Kuratowski offered the now-accepted definition[4][5] of the ordered pair (a, b):

(a, b)K := {{a}, {a, b}}.

Note that this definition is used even when the first and the second coordinates are identical:

(x, x)K = {{x}, {x, x}} = {{x}, {x}} = {{x}}

Given some ordered pair p, the property "x is the first coordinate of p" can be formulated as:

∀Y ∈p : x∈Y.

The property "x is the second coordinate of p" can be formulated as:

(∃Y ∈p : x∈Y ) ∧ (∀Y1, Y2∈p : Y1 = Y2 → (x/∈Y1 ∨ x/∈Y2)).

In the case that the left and right coordinates are identical, the right conjunct (∀Y1, Y2∈p : Y1 = Y2 → (x/∈Y1 ∨x/∈Y2)) is trivially true, since Y1 ≠ Y2 is never the case.This is how we can extract the first coordinate of a pair (using the notation for arbitrary intersection and arbitraryunion):

π1(p) =∪∩

p

This is how the second coordinate can be extracted:

π2(p) =∪

{x ∈∪

p |∪

p =∩

p → x /∈∩

p}

54 CHAPTER 8. ORDERED PAIR

Variants

The above Kuratowski definition of the ordered pair is “adequate” in that it satisfies the characteristic property that anordered pair must satisfy, namely that (a, b) = (x, y) ↔ (a = x) ∧ (b = y) . In particular, it adequately expresses'order', in that (a, b) = (b, a) is false unless b = a . There are other definitions, of similar or lesser complexity, thatare equally adequate:

• (a, b)reverse := {{b}, {a, b}};

• (a, b)short := {a, {a, b}};

• (a, b)01 := {{0, a}, {1, b}}. [6]

The reverse definition is merely a trivial variant of the Kuratowski definition, and as such is of no independent interest.The definition short is so-called because it requires two rather than three pairs of braces. Proving that short satisfiesthe characteristic property requires the Zermelo–Fraenkel set theory axiom of regularity.[7] Moreover, if one usesvon Neumann’s set-theoretic construction of the natural numbers, then 2 is defined as the set {0, 1} = {0, {0}}, whichis indistinguishable from the pair (0, 0) ₒᵣ . Yet another disadvantage of the short pair is the fact, that even if a andb are of the same type, the elements of the short pair are not. (However, if a = b then the short version keeps havingcardinality 2, which is something one might expect of any “pair”, including any “ordered pair”. Also note that theshort version is used in Tarski–Grothendieck set theory, upon which the Mizar system is founded.)

Proving that definitions satisfy the characteristic property

Prove: (a, b) = (c, d) if and only if a = c and b = d.Kuratowski:If. If a = c and b = d, then {{a}, {a, b}} = {{c}, {c, d}}. Thus (a, b)K = (c, d)K.Only if. Two cases: a = b, and a ≠ b.If a = b:

(a, b)K = {{a}, {a, b}} = {{a}, {a, a}} = {{a}}.(c, d)K = {{c}, {c, d}} = {{a}}.Thus {c} = {c, d} = {a}, which implies a = c and a = d. By hypothesis, a = b. Hence b = d.

If a ≠ b, then (a, b)K = (c, d)K implies {{a}, {a, b}} = {{c}, {c, d}}.

Suppose {c, d} = {a}. Then c = d = a, and so {{c}, {c, d}} = {{a}, {a, a}} = {{a}, {a}} = {{a}}. Butthen {{a}, {a, b}} would also equal {{a}}, so that b = a which contradicts a ≠ b.

Suppose {c} = {a, b}. Then a = b = c, which also contradicts a ≠ b.

Therefore {c} = {a}, so that c = a and {c, d} = {a, b}.

If d = a were true, then {c, d} = {a, a} = {a} ≠ {a, b}, a contradiction. Thus d = b is the case, so thata = c and b = d.

Reverse:(a, b)ᵣₑᵥₑᵣ ₑ = {{b}, {a, b}} = {{b}, {b, a}} = (b, a)K.If. If (a, b)ᵣₑᵥₑᵣ ₑ = (c, d)ᵣₑᵥₑᵣ ₑ, (b, a)K = (d, c)K. Therefore b = d and a = c.Only if. If a = c and b = d, then {{b}, {a, b}} = {{d}, {c, d}}. Thus (a, b)ᵣₑᵥₑᵣ ₑ = (c, d)ᵣₑᵥₑᵣ ₑ.Short:[8]

If: If a = c and b = d, then {a, {a, b}} = {c, {c, d}}. Thus (a, b) ₒᵣ = (c, d) ₒᵣ .Only if: Suppose {a, {a, b}} = {c, {c, d}}. Then a is in the left hand side, and thus in the right hand side. Becauseequal sets have equal elements, one of a = c or a = {c, d} must be the case.

8.2. DEFINING THE ORDERED PAIR USING SET THEORY 55

If a = {c, d}, then by similar reasoning as above, {a, b} is in the right hand side, so {a, b} = c or {a, b}= {c, d}.

If {a, b} = c then c is in {c, d} = a and a is in c, and this combination contradicts the axiomof regularity, as {a, c} has no minimal element under the relation “element of.”If {a, b} = {c, d}, then a is an element of a, from a = {c, d} = {a, b}, again contradictingregularity.

Hence a = c must hold.

Again, we see that {a, b} = c or {a, b} = {c, d}.

The option {a, b} = c and a = c implies that c is an element of c, contradicting regularity.So we have a = c and {a, b} = {c, d}, and so: {b} = {a, b} \ {a} = {c, d} \ {c} = {d}, so b = d.

8.2.4 Quine-Rosser definition

Rosser (1953)[9] employed a definition of the ordered pair due to Quine which requires a prior definition of the naturalnumbers. Let N be the set of natural numbers and x \ N be the elements of x not in N . Define

φ(x) = (x \ N) ∪ {n+ 1 : n ∈ (x ∩ N)}.

Applying this function simply increments every natural number in x. In particular, φ(x) does not contain the number0, so that for any sets x and y,

φ(x) = {0} ∪ φ(y).

Define the ordered pair (A, B) as

(A,B) = {φ(a) : a ∈ A} ∪ {φ(b) ∪ {0} : b ∈ B}.

Extracting all the elements of the pair that do not contain 0 and undoing φ yields A. Likewise, B can be recoveredfrom the elements of the pair that do contain 0.In type theory and in outgrowths thereof such as the axiomatic set theory NF, the Quine-Rosser pair has the sametype as its projections and hence is termed a “type-level” ordered pair. Hence this definition has the advantage ofenabling a function, defined as a set of ordered pairs, to have a type only 1 higher than the type of its arguments.This definition works only if the set of natural numbers is infinite. This is the case in NF, but not in type theory orin NFU. J. Barkley Rosser showed that the existence of such a type-level ordered pair (or even a “type-raising by 1”ordered pair) implies the axiom of infinity. For an extensive discussion of the ordered pair in the context of Quinianset theories, see Holmes (1998).[10]

8.2.5 Morse definition

Morse–Kelley set theory makes free use of proper classes.[11] Morse defined the ordered pair so that its projectionscould be proper classes as well as sets. (The Kuratowski definition does not allow this.) He first defined ordered pairswhose projections are sets in Kuratowski’s manner. He then redefined the pair

(x, y) = ({0} × s(x)) ∪ ({1} × s(y))

where the component Cartesian products are Kuratowski pairs of sets and where

s(x) = {∅} ∪ {{t}|t ∈ x}

56 CHAPTER 8. ORDERED PAIR

This renders possible pairs whose projections are proper classes. The Quine-Rosser definition above also admitsproper classes as projections. Similarly the triple is defined as a 3-tuple as follows:

(x, y, z) = ({0} × s(x)) ∪ ({1} × s(y)) ∪ ({2} × s(z))

The use of the singleton set s(x) which has an inserted empty set allows tuples to have the uniqueness property thatif a is an n-tuple and b is an m-tuple and a = b then n = m. Ordered triples which are defined as ordered pairs do nothave this property with respect to ordered pairs.

8.3 Category theory

A category-theoretic product A × B in a category of sets represents the set of ordered pairs, with the first elementcoming from A and the second coming from B. In this context the characteristic property above is a consequence ofthe universal property of the product and the fact that elements of a set X can be identified with morphisms from 1(a one element set) to X. While different objects may have the universal property, they are all naturally isomorphic.

8.4 References[1] Quine has argued that the set-theoretical implementations of the concept of the ordered pair is a paradigm for the clarifica-

tion of philosophical ideas (see "Word and Object", section 53). The general notion of such definitions or implementationsare discussed in Thomas Forster “Reasoning about theoretical entities”.

[2] Wiener’s paper “A Simplification of the logic of relations” is reprinted, together with a valuable commentary on pages224ff in van Heijenoort, Jean (1967), From Frege to Gödel: A Source Book in Mathematical Logic, 1979-1931, HarvardUniversity Press, Cambridge MA, ISBN 0-674-32449-8 (pbk.). van Heijenoort states the simplification this way: “Bygiving a definition of the ordered pair of two elements in terms of class operations, the note reduced the theory of relationsto that of classes”.

[3] cf introduction to Wiener’s paper in van Heijenoort 1967:224

[4] cf introduction to Wiener’s paper in van Heijenoort 1967:224. van Heijenoort observes that the resulting set that representsthe ordered pair “has a type higher by 2 than the elements (when they are of the same type)"; he offers references that showhow, under certain circumstances, the type can be reduced to 1 or 0.

[5] Kuratowski, Casimir (1921). “Sur la notion de l'ordre dans la Théorie des Ensembles” (PDF). Fundamenta Mathematicae2 (1): 161–171.

[6] This differs from Hausdorff’s definition in not requiring the two elements 0 and 1 to be distinct from a and b.

[7] Tourlakis, George (2003) Lectures in Logic and Set Theory. Vol. 2: Set Theory. Cambridge Univ. Press. PropositionIII.10.1.

[8] For a formal Metamath proof of the adequacy of short, see here (opthreg). Also see Tourlakis (2003), Proposition III.10.1.

[9] J. Barkley Rosser, 1953. Logic for Mathematicians. McGraw-Hill.

[10] Holmes, Randall (1998) Elementary Set Theory with a Universal Set. Academia-Bruylant. The publisher has graciouslyconsented to permit diffusion of this monograph via the web.

[11] Morse, Anthony P. (1965). A Theory of Sets. Academic Press.

Chapter 9

Orientation (graph theory)

In graph theory, an orientation of an undirected graph is an assignment of a direction to each edge, turning the initialgraph into a directed graph.

9.1 Oriented graphs

A directed graph is called an oriented graph if it is the orientation of an undirected graph. Among directed graphs,the oriented graphs are the ones that have no self-loops, no multiple adjacencies, and no 2-cycles.[1]

A tournament is an orientation of a complete graph. A polytree is an orientation of an undirected tree.[2] Sumner’sconjecture states that every tournament with 2n − 2 vertices contains every polytree with n vertices.[3]

The number of non-isomorphic oriented graphs with n vertices (for n = 1, 2, 3, ...) is

1, 2, 7, 42, 582, 21480, 2142288, 575016219, 415939243032, ... (sequence A001174 in OEIS).

Oriented graphs are in one-to-one correspondence with complete directed graphs (graphs in which there is a directededge in one or both directions between every pair of distinct vertices). A complete directed graph can be converted toan oriented graph by removing every 2-cycle, and conversely an oriented graph can be converted to a complete directedgraph by adding a 2-cycle between every pair of vertices that are not endpoints of an edge; these correspondencesare bijective. Therefore, the same sequence of numbers also solves the graph enumeration problem for completedigraphs. There is an explicit but complicated formula for the numbers in this sequence.[4]

9.2 Constrained orientations

A strong orientation is an orientation that results in a strongly connected graph. The closely related totally cyclicorientations are orientations in which every edge belongs to at least one simple cycle. An orientation of an undirectedgraph G is totally cyclic if and only if it is a strong orientation of every connected component of G. Robbins’ theoremstates that a graph has a strong orientation if and only if it is 2-edge-connected; disconnected graphs may have totallycyclic orientations, but only if they have no bridges.[5]

An acyclic orientation is an orientation that results in a directed acyclic graph. Every graph has an acyclic orientation;all acyclic orientations may be obtained by placing the vertices into a sequence, and then directing each edge fromthe earlier of its endpoints in the sequence to the later endpoint. The Gallai–Hasse–Roy–Vitaver theorem states thata graph has an acyclic orientation in which the longest path has at most k vertices if and only if it can be colored withat most k colors.[6] Acyclic orientations and totally cyclic orientations are related to each other by planar duality. Anacyclic orientation with a single source and a single sink is called a bipolar orientation.[7]

A transitive orientation is an orientation such that the resulting directed graph is its own transitive closure. Thegraphs with transitive orientations are called comparability graphs; they may be defined from a partially ordered setby making two elements adjacent whenever they are comparable in the partial order.[8] A transitive orientation, if oneexists, can be found in linear time.[9] However, testing whether the resulting orientation (or any given orientation) isactually transitive requires more time, as it is equivalent in complexity to matrix multiplication.

57

58 CHAPTER 9. ORIENTATION (GRAPH THEORY)

An Eulerian orientation of an undirected graph is an orientation in which each vertex has equal in-degree and out-degree. Eulerian orientations of grid graphs arise in statistical mechanics in the theory of ice-type models.[10]

9.3 References[1] Diestel, Reinhard (2005), “1.10 Other notions of graphs”, Graph Theory (PDF) (3rd ed.), Springer, ISBN 3-540-26182-6.

[2] Rebane, George; Pearl, Judea (1987), “The recovery of causal poly-trees from statistical data”, in Proc. 3rd Annual Con-ference on Uncertainty in Artificial Intelligence (UAI 1987), Seattle, WA, USA, July 1987 (PDF), pp. 222–228.

[3] Sumner’s Universal Tournament Conjecture, Douglas B. West, retrieved 2012-08-02.

[4] Harary, Frank; Palmer, Edgar M. (1973), “Formula 5.4.13”, Graphical Enumeration, New York: Academic Press, p. 133,MR 0357214.

[5] Robbins, H. E. (1939), “A theorem on graphs, with an application to a problem of traffic control”, The American Mathe-matical Monthly 46: 281–283, doi:10.2307/2303897.

[6] Nešetřil, Jaroslav; Ossona de Mendez, Patrice (2012), “Theorem 3.13”, Sparsity: Graphs, Structures, and Algorithms, Algo-rithms and Combinatorics 28, Heidelberg: Springer, p. 42, doi:10.1007/978-3-642-27875-4, ISBN 978-3-642-27874-7,MR 2920058.

[7] de Fraysseix, Hubert; de Mendez, Patrice Ossona; Rosenstiehl, Pierre (1995), “Bipolar orientations revisited”, DiscreteApplied Mathematics 56 (2-3): 157–179, doi:10.1016/0166-218X(94)00085-R, MR 1318743.

[8] Ghouila-Houri, Alain (1962), “Caractérisation des graphes non orientés dont on peut orienter les arrêtes de manière àobtenir le graphe d'une relation d'ordre”, Les Comptes rendus de l'Académie des sciences 254: 1370–1371, MR 0172275.

[9] McConnell, R. M.; Spinrad, J. (1997), “Linear-time transitive orientation”, 8th ACM-SIAM Symposium on Discrete Algo-rithms, pp. 19–25.

[10] Mihail, Milena; Winkler, Peter (1992), “On the Number of Eularian Orientations of a Graph”, Proceedings of the ThirdAnnual ACM-SIAM Symposium on Discrete Algorithms, SODA '92, Philadelphia, PA, USA: Society for Industrial andApplied Mathematics, pp. 138–145, ISBN 0-89791-466-X.

9.4 External links• Weisstein, Eric W., “Graph Orientation”, MathWorld.

• Weisstein, Eric W., “Oriented Graph”, MathWorld.

Chapter 10

Path (graph theory)

For the family of graphs known as paths, see Path graph.In graph theory, a path in a graph is a finite or infinite sequence of edges which connect a sequence of vertices

110 111

100 101

000

010

001

011

A hypercube graph showing a Hamiltonian path in red, and a longest induced path in bold black.

which, by most definitions, are all distinct from one another. In a directed graph, a directed path (sometimes calleddipath[1]) is again a sequence of edges (or arcs) which connect a sequence of vertices, but with the added restrictionthat the edges all be directed in the same direction.

59

60 CHAPTER 10. PATH (GRAPH THEORY)

Paths are fundamental concepts of graph theory, described in the introductory sections of most graph theory texts.See e.g. Bondy and Murty (1976), Gibbons (1985), or Diestel (2005). Korte et al. (1990) cover more advancedalgorithmic topics concerning paths in graphs.

10.1 Definitions

A path is a trail in which all vertices (except possibly the first and last) are distinct. A trail is a walk in which all edgesare distinct. Awalk of length k in a graph is an alternating sequence of vertices and edges, v0, e0, v1, e1, v2, . . . , vk−1, ek−1, vk, which begins and ends with vertices. If the graph is directed, then ei is an arc from vi to vi+1 . An infinite path isan alternating sequence of the same type described here, but with no first or last vertex, and a semi-infinite path (alsoray) has a first vertex, v0 , but no last vertex. Most authors require that all of the edges and vertices be distinct fromone another.A weighted graph associates a value (weight) with every edge in the graph. The weight of a path in a weighted graphis the sum of the weights of the traversed edges. Sometimes the words cost or length are used instead of weight.

10.2 Examples• A graph is connected if there are paths containing each pair of vertices.

• A directed graph is strongly connected if there are oppositely oriented directed paths containing each pair ofvertices.

• A path such that no graph edges connect two nonconsecutive path vertices is called an induced path.

• A path that includes every vertex of the graph is known as a Hamiltonian path.

• Two paths are vertex-independent (alternatively, internally vertex-disjoint) if they do not have any internal vertexin common. Similarly, two paths are edge-independent (or edge-disjoint) if they do not have any internal edgein common. Two internally vertex-disjoint paths are edge-disjoint, but the converse is not necessarily true.

• The distance between two vertices in a graph is the length of a shortest path between them, if one exists, andotherwise the distance is infinity.

• The diameter of a connected graph is the largest distance (defined above) between pairs of vertices of thegraph.

Several algorithms exist to find shortest and longest paths in graphs, with the important distinction that the formerproblem is computationally much easier than the latter, unless P=NP. Dijkstra’s algorithm produces a list of shortestpaths from a source vertex to every other vertex in directed and undirected graphs with non-negative edge weights (orno edge weights), whilst the Bellman–Ford algorithm can be applied to directed graphs with negative edge weights.The Floyd–Warshall algorithm can be used to find the shortest paths between all pairs of vertices in weighted directedgraphs.

10.3 See also• Glossary of graph theory

• Path graph

• Polygonal chain

• Shortest path problem

• Longest path problem

• Dijkstra’s algorithm

• Bellman–Ford algorithm

10.4. REFERENCES 61

• Floyd–Warshall algorithm

• Self-avoiding walk

10.4 References[1] Graph Structure Theory: Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on GraphMinors, Held

June 22 to July 5, 1991,, p.205

• Bondy, J. A.; Murty, U. S. R. (1976). Graph Theory with Applications. North Holland. pp. 12–21. ISBN0-444-19451-7.

• Diestel, Reinhard (2005). Graph Theory (3rd ed. ed.). Graduate Texts in Mathematics, vol. 173, Springer-Verlag. pp. 6–9. ISBN 3-540-26182-6.

• Gibbons, A. (1985). Algorithmic Graph Theory. Cambridge University Press. pp. 5–6. ISBN 0-521-28881-9.

• Korte, Bernhard; Lovász, László; Prömel, Hans Jürgen; Schrijver, Alexander (Eds.) (1990). Paths, Flows, andVLSI-Layout. Algorithms and Combinatorics 9, Springer-Verlag. ISBN 0-387-52685-4.

Chapter 11

Set (mathematics)

This article is about what mathematicians call “intuitive” or “naive” set theory. For a more detailed account, see Naiveset theory. For a rigorous modern axiomatic treatment of sets, see Set theory.In mathematics, a set is a collection of distinct objects, considered as an object in its own right. For example,

A set of polygons in a Venn diagram

the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectivelythey form a single set of size three, written {2,4,6}. Sets are one of the most fundamental concepts in mathematics.Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as afoundation from which nearly all of mathematics can be derived. In mathematics education, elementary topics suchas Venn diagrams are taught at a young age, while more advanced concepts are taught as part of a university degree.

62

11.1. DEFINITION 63

The German word Menge, rendered as “set” in English, was coined by Bernard Bolzano in his work The Paradoxesof the Infinite.

11.1 Definition

Passage with the original set definition of Georg Cantor

A set is a well defined collection of distinct objects. The objects that make up a set (also known as the elements ormembers of a set) can be anything: numbers, people, letters of the alphabet, other sets, and so on. Georg Cantor,the founder of set theory, gave the following definition of a set at the beginning of his Beiträge zur Begründung dertransfiniten Mengenlehre:[1]

A set is a gathering together into a whole of definite, distinct objects of our perception [Anschauung]or of our thought—which are called elements of the set.

Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the sameelements.[2]

There is the image popular, that sets are like boxes containing their elements. But there is a huge difference betweenboxes and sets. While boxes don't change their identity when objects are removed from or added to them, sets changetheir identity when their elements change. So its better to have the image of a set as the content of an imaginary box:

• A set of polygons

• The same set as a box

• The set as the content of a box

Cantor’s definition turned out to be inadequate for formal mathematics; instead, the notion of a “set” is taken as anundefined primitive in axiomatic set theory, and its properties are defined by the Zermelo–Fraenkel axioms. Themost basic properties are that a set has elements, and that two sets are equal (one and the same) if and only if everyelement of each set is an element of the other.

11.2 Describing sets

Main article: Set notation

There are two ways of describing, or specifying the members of, a set. One way is by intensional definition, using arule or semantic description:

64 CHAPTER 11. SET (MATHEMATICS)

A is the set whose members are the first four positive integers.B is the set of colors of the French flag.

The second way is by extension – that is, listing each member of the set. An extensional definition is denoted byenclosing the list of members in curly brackets:

C = {4, 2, 1, 3}D = {blue, white, red}.

One often has the choice of specifying a set either intensionally or extensionally. In the examples above, for instance,A = C and B = D.There are two important points to note about sets. First, a set can have two or more members which are identical, forexample, {11, 6, 6}. However, we say that two sets which differ only in that one has duplicate members are in factexactly identical (see Axiom of extensionality). Hence, the set {11, 6, 6} is exactly identical to the set {11, 6}. Thesecond important point is that the order in which the elements of a set are listed is irrelevant (unlike for a sequenceor tuple). We can illustrate these two important points with an example:

{6, 11} = {11, 6} = {11, 6, 6, 11} .

For sets with many elements, the enumeration of members can be abbreviated. For instance, the set of the firstthousand positive integers may be specified extensionally as

{1, 2, 3, ..., 1000},

where the ellipsis ("...”) indicates that the list continues in the obvious way. Ellipses may also be used where sets haveinfinitely many members. Thus the set of positive even numbers can be written as {2, 4, 6, 8, ... }.The notation with braces may also be used in an intensional specification of a set. In this usage, the braces have themeaning “the set of all ...”. So, E = {playing card suits} is the set whose four members are ♠, ♦, ♥, and ♣. A moregeneral form of this is set-builder notation, through which, for instance, the set F of the twenty smallest integers thatare four less than perfect squares can be denoted

F = {n2 − 4 : n is an integer; and 0 ≤ n ≤ 19}.

In this notation, the colon (":") means “such that”, and the description can be interpreted as "F is the set of all numbersof the form n2 − 4, such that n is a whole number in the range from 0 to 19 inclusive.” Sometimes the vertical bar("|") is used instead of the colon.

11.3 Membership

Main article: Element (mathematics)

If B is a set and x is one of the objects of B, this is denoted x ∈ B, and is read as “x belongs to B”, or “x is an elementof B”. If y is not a member of B then this is written as y ∉ B, and is read as “y does not belong to B”.For example, with respect to the sets A = {1,2,3,4}, B = {blue, white, red}, and F = {n2 − 4 : n is an integer; and 0≤ n ≤ 19} defined above,

4 ∈ A and 12 ∈ F; but9 ∉ F and green ∉ B.

11.3. MEMBERSHIP 65

11.3.1 Subsets

Main article: Subset

If every member of set A is also a member of set B, then A is said to be a subset of B, written A ⊆ B (also pronouncedA is contained in B). Equivalently, we can write B ⊇ A, read as B is a superset of A, B includes A, or B contains A. Therelationship between sets established by ⊆ is called inclusion or containment.If A is a subset of, but not equal to, B, then A is called a proper subset of B, written A ⊊ B (A is a proper subset of B)or B ⊋ A (B is a proper superset of A).Note that the expressions A ⊂ B and B ⊃ A are used differently by different authors; some authors use them to meanthe same as A ⊆ B (respectively B ⊇ A), whereas other use them to mean the same as A ⊊ B (respectively B ⊋ A).

AB

A is a subset of B

Example:

• The set of all men is a proper subset of the set of all people.• {1, 3} ⊆ {1, 2, 3, 4}.• {1, 2, 3, 4} ⊆ {1, 2, 3, 4}.

66 CHAPTER 11. SET (MATHEMATICS)

The empty set is a subset of every set and every set is a subset of itself:

• ∅ ⊆ A.• A ⊆ A.

An obvious but useful identity, which can often be used to show that two seemingly different sets are equal:

• A = B if and only if A ⊆ B and B ⊆ A.

A partition of a set S is a set of nonempty subsets of S such that every element x in S is in exactly one of these subsets.

11.3.2 Power sets

Main article: Power set

The power set of a set S is the set of all subsets of S. Note that the power set contains S itself and the empty setbecause these are both subsets of S. For example, the power set of the set {1, 2, 3} is {{1, 2, 3}, {1, 2}, {1, 3}, {2,3}, {1}, {2}, {3}, ∅}. The power set of a set S is usually written as P(S).The power set of a finite set with n elements has 2n elements. This relationship is one of the reasons for the terminologypower set. For example, the set {1, 2, 3} contains three elements, and the power set shown above contains 23 = 8elements.The power set of an infinite (either countable or uncountable) set is always uncountable. Moreover, the power set ofa set is always strictly “bigger” than the original set in the sense that there is no way to pair every element of S withexactly one element of P(S). (There is never an onto map or surjection from S onto P(S).)Every partition of a set S is a subset of the powerset of S.

11.4 Cardinality

Main article: Cardinality

The cardinality | S | of a set S is “the number of members of S.” For example, if B = {blue, white, red}, | B | = 3.There is a unique set with no members and zero cardinality, which is called the empty set (or the null set) and isdenoted by the symbol ∅ (other notations are used; see empty set). For example, the set of all three-sided squares haszero members and thus is the empty set. Though it may seem trivial, the empty set, like the number zero, is importantin mathematics; indeed, the existence of this set is one of the fundamental concepts of axiomatic set theory.Some sets have infinite cardinality. The set N of natural numbers, for instance, is infinite. Some infinite cardinalitiesare greater than others. For instance, the set of real numbers has greater cardinality than the set of natural numbers.However, it can be shown that the cardinality of (which is to say, the number of points on) a straight line is the sameas the cardinality of any segment of that line, of the entire plane, and indeed of any finite-dimensional Euclideanspace.

11.5 Special sets

There are some sets that hold great mathematical importance and are referred to with such regularity that they haveacquired special names and notational conventions to identify them. One of these is the empty set, denoted {} or ∅.Another is the unit set {x}, which contains exactly one element, namely x.[2] Many of these sets are represented usingblackboard bold or bold typeface. Special sets of numbers include

• P or ℙ, denoting the set of all primes: P = {2, 3, 5, 7, 11, 13, 17, ...}.

• N or ℕ, denoting the set of all natural numbers: N = {1, 2, 3, . . .} (sometimes defined containing 0).

11.6. BASIC OPERATIONS 67

• Z or ℤ, denoting the set of all integers (whether positive, negative or zero): Z = {..., −2, −1, 0, 1, 2, ...}.

• Q or ℚ, denoting the set of all rational numbers (that is, the set of all proper and improper fractions): Q = {a/b: a, b ∈ Z, b ≠ 0}. For example, 1/4 ∈ Q and 11/6 ∈ Q. All integers are in this set since every integer a can beexpressed as the fraction a/1 (Z ⊊ Q).

• R or ℝ, denoting the set of all real numbers. This set includes all rational numbers, together with all irrationalnumbers (that is, numbers that cannot be rewritten as fractions, such as √2, as well as transcendental numberssuch as π, e and numbers that cannot be defined).

• C or ℂ, denoting the set of all complex numbers: C = {a + bi : a, b ∈ R}. For example, 1 + 2i ∈ C.

• H or ℍ, denoting the set of all quaternions: H = {a + bi + cj + dk : a, b, c, d ∈ R}. For example, 1 + i + 2j −k ∈ H.

Positive and negative sets are denoted by a superscript - or +. For example ℚ+ represents the set of positive rationalnumbers.Each of the above sets of numbers has an infinite number of elements, and each can be considered to be a propersubset of the sets listed below it. The primes are used less frequently than the others outside of number theory andrelated fields.

11.6 Basic operations

There are several fundamental operations for constructing new sets from given sets.

11.6.1 Unions

The union of A and B, denoted A ∪ B

68 CHAPTER 11. SET (MATHEMATICS)

Main article: Union (set theory)

Two sets can be “added” together. The union of A and B, denoted by A ∪ B, is the set of all things that are membersof either A or B.Examples:

• {1, 2} ∪ {1, 2} = {1, 2}.• {1, 2} ∪ {2, 3} = {1, 2, 3}.• {1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}

Some basic properties of unions:

• A ∪ B = B ∪ A.• A ∪ (B ∪ C) = (A ∪ B) ∪ C.• A ⊆ (A ∪ B).• A ∪ A = A.• A ∪ ∅ = A.• A ⊆ B if and only if A ∪ B = B.

11.6.2 Intersections

Main article: Intersection (set theory)

A new set can also be constructed by determining which members two sets have “in common”. The intersection of Aand B, denoted by A ∩ B, is the set of all things that are members of both A and B. If A ∩ B = ∅, then A and B aresaid to be disjoint.Examples:

• {1, 2} ∩ {1, 2} = {1, 2}.• {1, 2} ∩ {2, 3} = {2}.

Some basic properties of intersections:

• A ∩ B = B ∩ A.• A ∩ (B ∩ C) = (A ∩ B) ∩ C.• A ∩ B ⊆ A.• A ∩ A = A.• A ∩ ∅ = ∅.• A ⊆ B if and only if A ∩ B = A.

11.6.3 Complements

Main article: Complement (set theory)

Two sets can also be “subtracted”. The relative complement of B in A (also called the set-theoretic difference of A andB), denoted by A \ B (or A − B), is the set of all elements that are members of A but not members of B. Note that itis valid to “subtract” members of a set that are not in the set, such as removing the element green from the set {1, 2,3}; doing so has no effect.In certain settings all sets under discussion are considered to be subsets of a given universal set U. In such cases, U \A is called the absolute complement or simply complement of A, and is denoted by A′.Examples:

11.6. BASIC OPERATIONS 69

The intersection of A and B, denoted A ∩ B.

• {1, 2} \ {1, 2} = ∅.

• {1, 2, 3, 4} \ {1, 3} = {2, 4}.

• If U is the set of integers, E is the set of even integers, and O is the set of odd integers, then U \ E= E′ = O.

Some basic properties of complements:

• A \ B ≠ B \ A for A ≠ B.

• A ∪ A′ = U.

• A ∩ A′ = ∅.

• (A′)′ = A.

• A \ A = ∅.

• U′ = ∅ and ∅′ = U.

• A \ B = A ∩ B′.

An extension of the complement is the symmetric difference, defined for sets A, B as

A∆B = (A \B) ∪ (B \A).

For example, the symmetric difference of {7,8,9,10} and {9,10,11,12} is the set {7,8,11,12}.

70 CHAPTER 11. SET (MATHEMATICS)

The relative complementof B in A

11.6.4 Cartesian product

Main article: Cartesian product

A new set can be constructed by associating every element of one set with every element of another set. The Cartesianproduct of two sets A and B, denoted by A × B is the set of all ordered pairs (a, b) such that a is a member of A andb is a member of B.Examples:

• {1, 2} × {red, white} = {(1, red), (1, white), (2, red), (2, white)}.• {1, 2} × {red, white, green} = {(1, red), (1, white), (1, green), (2, red), (2, white), (2, green) }.• {1, 2} × {1, 2} = {(1, 1), (1, 2), (2, 1), (2, 2)}.

Some basic properties of cartesian products:

• A × ∅ = ∅.• A × (B ∪ C) = (A × B) ∪ (A × C).• (A ∪ B) × C = (A × C) ∪ (B × C).

Let A and B be finite sets. Then

• | A × B | = | B × A | = | A | × | B |.

For example,

• {a,b,c}×{d,e,f}={(a,d),(a,e),(a,f),(b,d),(b,e),(b,f),(c,d),(c,e),(c,f)}.

11.7. APPLICATIONS 71

The complement of A in U

11.7 Applications

Set theory is seen as the foundation from which virtually all of mathematics can be derived. For example, structuresin abstract algebra, such as groups, fields and rings, are sets closed under one or more operations.One of the main applications of naive set theory is constructing relations. A relation from a domain A to a codomainB is a subset of the Cartesian product A × B. Given this concept, we are quick to see that the set F of all ordered pairs(x, x2), where x is real, is quite familiar. It has a domain set R and a codomain set that is also R, because the set of allsquares is subset of the set of all reals. If placed in functional notation, this relation becomes f(x) = x2. The reasonthese two are equivalent is for any given value, y that the function is defined for, its corresponding ordered pair, (y,y2) is a member of the set F.

11.8 Axiomatic set theory

Main article: Axiomatic set theory

Although initially naive set theory, which defines a set merely as any well-defined collection, was well accepted, itsoon ran into several obstacles. It was found that this definition spawned several paradoxes, most notably:

• Russell’s paradox—It shows that the “set of all sets that do not contain themselves,” i.e. the “set” { x : x is a setand x ∉ x } does not exist.

• Cantor’s paradox—It shows that “the set of all sets” cannot exist.

The reason is that the phrase well-defined is not very well defined. It was important to free set theory of theseparadoxes because nearly all of mathematics was being redefined in terms of set theory. In an attempt to avoid theseparadoxes, set theory was axiomatized based on first-order logic, and thus axiomatic set theory was born.

72 CHAPTER 11. SET (MATHEMATICS)

The symmetric difference of A and B

For most purposes however, naive set theory is still useful.

11.9 Principle of inclusion and exclusion

Main article: Inclusion-exclusion principle

This principle gives us the cardinality of the union of sets.

|A1 ∪A2 ∪A3 ∪ . . . ∪An| =(|A1|+ |A2|+ |A3|+ . . . |An|)−(|A1 ∩A2|+ |A1 ∩A3|+ . . . |An−1 ∩An|)+. . .+

(−1)n−1

(|A1 ∩A2 ∩A3 ∩ . . . ∩An|)

11.10 De Morgan’s Law

De Morgan stated two laws about Sets.If A and B are any two Sets then,

• (A ∪ B)′ = A′ ∩ B′

The complement of A union B equals the complement of A intersected with the complement of B.

• (A ∩ B)′ = A′ ∪ B′

The complement of A intersected with B is equal to the complement of A union to the complement of B.

11.11. SEE ALSO 73

11.11 See also• Set notation

• Mathematical object

• Alternative set theory

• Axiomatic set theory

• Category of sets

• Class (set theory)

• Dense set

• Family of sets

• Fuzzy set

• Internal set

• Mereology

• Multiset

• Naive set theory

• Principia Mathematica

• Rough set

• Russell’s paradox

• Sequence (mathematics)

• Taxonomy

• Tuple

11.12 Notes[1] “EineMenge, ist die Zusammenfassung bestimmter, wohlunterschiedenerObjekte unserer Anschauung oder unseresDenkens

– welche Elemente der Menge genannt werden – zu einem Ganzen.”

[2] Stoll, Robert. Sets, Logic and Axiomatic Theories. W. H. Freeman and Company. p. 5.

11.13 References• Dauben, JosephW., Georg Cantor: His Mathematics and Philosophy of the Infinite, Boston: Harvard UniversityPress (1979) ISBN 978-0-691-02447-9.

• Halmos, Paul R., Naive Set Theory, Princeton, N.J.: Van Nostrand (1960) ISBN 0-387-90092-6.

• Stoll, Robert R., Set Theory and Logic, Mineola, N.Y.: Dover Publications (1979) ISBN 0-486-63829-4.

• Velleman, Daniel, How To Prove It: A Structured Approach, Cambridge University Press (2006) ISBN 978-0-521-67599-4

11.14 External links• C2 Wiki – Examples of set operations using English operators.

• Mathematical Sets: Elements, Intersections & Unions, Education Portal Academy

Chapter 12

Unordered pair

In mathematics, an unordered pair or pair set is a set of the form {a, b}, i.e. a set having two elements a and b withno particular relation between them. In contrast, an ordered pair (a, b) has a as its first element and b as its secondelement.While the two elements of an ordered pair (a, b) need not be distinct, modern authors only call {a, b} an unorderedpair if a ≠ b.[1] But for a few authors a singleton is also considered an unordered pair, although today, most would saythat {a,a} is a multiset. It is typical to use the term unordered pair even in the situation where the elements a and bcould be equal, as long as this equality has not yet been established.A set with precisely two elements is also called a 2-set or (rarely) a binary set.An unordered pair is a finite set; its cardinality (number of elements) is 2 or (if the two elements are not distinct) 1.In axiomatic set theory, the existence of unordered pairs is required by an axiom, the axiom of pairing.More generally, an unordered n-tuple is a set of the form {a1, a2,... an}.[2]

12.1 Notes[1] Düntsch, Ivo; Gediga, Günther (2000), Sets, Relations, Functions, Primers Series, Methodos, ISBN 978-1-903280-00-3.

Fraenkel, Adolf (1928), Einleitung in die Mengenlehre, Berlin, New York: Springer-Verlag.Roitman, Judith (1990), Introduction to modern set theory, New York: John Wiley & Sons, ISBN 978-0-471-63519-2.Schimmerling, Ernest (2008), Undergraduate set theory.

[2] Hrbacek, Karel; Jech, Thomas (1999), Introduction to set theory (3rd ed.), New York: Dekker, ISBN 978-0-8247-7915-3.Rubin, Jean E. (1967), Set theory for the mathematician, Holden-Day.Takeuti, Gaisi; Zaring, Wilson M. (1971), Introduction to axiomatic set theory, Graduate Texts in Mathematics, Berlin,New York: Springer-Verlag.

12.2 References• Enderton, Herbert (1977), Elements of set theory, Boston, MA: Academic Press, ISBN 978-0-12-238440-0.

74

12.3. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 75

12.3 Text and image sources, contributors, and licenses

12.3.1 Text• Adjacencymatrix Source: https://en.wikipedia.org/wiki/Adjacency_matrix?oldid=652744218Contributors: AxelBoldt, Tbackstr, Tomo,

Michael Hardy, Booyabazooka, TakuyaMurata, Mbogelund, Schneelocke, Dcoetzee, Dysprosia, Reina riemann, Aleph4, Fredrik, Math-Martin, Bkell, ElBenevolente, Giftlite, BenFrantzDale, Arthouse~enwiki, MarkSweep, Tomruen, Abdull, Rich Farmbrough, Bender235,Zaslav, Gauge, Rgdboer, Phils, Burn, Cburnett, Oleg Alexandrov, Timendum, YurikBot, Jpbowen, Jogers, Sneftel, SmackBot, Ttzz,Kneufeld, Senfo, Abha Jain, Tamfang, Juffi, EVula, Dreadstar, Mdrine, Paulish, Tim Q. Wells, Beetstra, Hu12, Rhetth, CmdrObot,Only2sea, Thijs!bot, Headbomb, Olenielsen, Salgueiro~enwiki, Natelewis, Xeno, JamesBWatson, David Eppstein, Yonidebot, Squids andChips, VolkovBot, LokiClock, YuryKirienko, Morre~enwiki, Periergeia, JackSchmidt, Garyzx, Watchduck, Bender2k14, Yoav HaCo-hen, Addbot, Fgnievinski, LaaknorBot, Debresser, Matěj Grabovský, ,سعی Luckas-bot, Calle, Milk’s Favorite Bot II, AnomieBOT, Twri,ArthurBot, SPTWriter, Miym, Bitsianshash, X7q, Jean Honorio, Sławomir Biały, Chenopodiaceous, RobinK, TobeBot, Patmorin, John ofReading, WikitanvirBot, Felix Hoffmann, ZéroBot, ChuispastonBot, Wcherowi, Helpful Pixie Bot, Shilpi4560, Snowcream, DarafshBot,Kompot 3 and Anonymous: 73

• Directed graph Source: https://en.wikipedia.org/wiki/Directed_graph?oldid=667034926 Contributors: Michael Hardy, Booyabazooka,Altenmann, Bkell, Giftlite, Vadmium, Andreas Kaufmann, Zaslav, Grue, Linas, SixWingedSeraph, Chobot, Wavelength, Stepa, Wook-ieInHeat, BiT, Nbarth, Delcypher, Meno25, Was a bee, Kozuch, Headbomb, Hamaryns, JAnDbot, Catgut, David Eppstein, R'n'B, Mar-cuse7~enwiki, M-le-mot-dit, Llorenzi, VolkovBot, HughD, Constant314, TXiKiBoT, Shauncutts, PaulTanenbaum, Rinix, Henry Delforn(old), Justin W Smith, JP.Martin-Flatin, Brews ohare, NuclearWarfare, MystBot, Addbot, Jarble, Luckas-bot, Ptbotgourou, Pcap, Calle,Dzied Bulbash, Bryan.burgers, Twri, Ms.wiki.us, Xqbot, ,آرمان Anne Bauval, Miym, Corruptcopper, Einkil, Mark Renier, Ricardo Fer-reira de Oliveira, Pierre5018, EmausBot, WikitanvirBot, Sinuhe20, ChuispastonBot, Joerg Bader, Helpful Pixie Bot, Alesiom, Markviking, Saranavan, Werddemer, SofjaKovalevskaja and Anonymous: 36

• Element (mathematics) Source: https://en.wikipedia.org/wiki/Element_(mathematics)?oldid=666614530 Contributors: Michael Hardy,Charles Matthews, Hyacinth, Chuunen Baka, Tobias Bergemann, Giftlite, Dbenbenn, Maximaximax, Eep², Brianjd, Dissipate, Paul Au-gust, Elwikipedista~enwiki, Nickj, Mlm42, Oleg Alexandrov, Mutford, Rjwilmsi, Salix alba, Chobot, YurikBot, RobotE, Thane, Goffrie,InverseHypercube, Mhss, Octahedron80, Gracenotes, Khoikhoi, MichaelBillington, Wvbailey, Bjankuloski06en~enwiki, SpyMagician,16@r, LimWei Quan, Mets501, JohnCD, Gregbard, Cydebot, Epbr123, Enlil2, Zorro CX, KeypadSDM, Ttwo, Idioma-bot, Pleasantville,SieBot, Paolo.dL, Harry~enwiki, ClueBot, Phileasson, Heckledpie, SoxBot III, RMFan1, Addbot, Professor Calculus, Betterusername,Binary TSO, Ben Ben, Yobot, TaBOT-zerem, KamikazeBot, Ciphers, Zach2231, Materialscientist, Citation bot, ArthurBot, DSisyphBot,FrescoBot, AllCluesKey, Spartan S58, Pinethicket, Jauhienij, Jophos, Rsagira58, EmausBot, Solarra, Traxs7, Ebrambot, Chuispaston-Bot, ClueBot NG, Wcherowi, SusikMkr, Atomician, Wikih101, Funnydiamond99, Aciganj, YFdyh-bot, Saehry, Stephan Kulla, Jwall42,Sonicbethesame and Anonymous: 78

• Glossary of graph theory Source: https://en.wikipedia.org/wiki/Glossary_of_graph_theory?oldid=666492791 Contributors: DamianYerrick, XJaM, Nonenmac, Tomo, Edward, Patrick, Michael Hardy, Wshun, Booyabazooka, Dcljr, TakuyaMurata, GTBacchus, Eric119,CharlesMatthews, Dcoetzee, Dysprosia, Doradus, Reina riemann, Markhurd, Maximus Rex, Hyacinth, Populus, Altenmann, MathMartin,Bkell, Giftlite, Dbenbenn, Brona, Sundar, GGordonWorleyIII, HorsePunchKid, Peter Kwok, D6, Rich Farmbrough, ArnoldReinhold, PaulAugust, Bender235, Zaslav, Kjoonlee, Elwikipedista~enwiki, El C, Yitzhak, TheSolomon, A1kmm, 3mta3, Jérôme, Ricky81682, Rd-vdijk, Oleg Alexandrov, Joriki, Linas, MattGiuca, Ruud Koot, Jwanders, Xiong, Lasunncty, SixWingedSeraph, Grammarbot, Tizio, Salixalba, Mathbot, Margosbot~enwiki, Sunayana, Pojo, Quuxplusone, Vonkje, N8wilson, Chobot, Algebraist, YurikBot, Me and, Altoid,Grubber, Archelon, Gaius Cornelius, Rick Norwood, Ott2, Closedmouth, SmackBot, Stux, Achab, Brick Thrower, Mgreenbe, Mcld,[email protected], Lansey, Thechao, JLeander, DVanDyck, Quaeler, RekishiEJ, CmdrObot, Csaracho, Citrus538, Jokes Free4Me,Cydebot, Starcrab, Quintopia, Ferris37, Scarpy, Headbomb, Salgueiro~enwiki, Spanningtree, David Eppstein, JoergenB, Kope, Copy-ToWiktionaryBot, R'n'B, Leyo, Mikhail Dvorkin, The Transliterator, Ratfox, MentorMentorum, Skaraoke, SanitySolipsism, AnonymousDissident, PaulTanenbaum, Ivan Štambuk, Whorush, Eggwadi, Thehotelambush, Doc honcho, Anchor Link Bot, Rsdetsch, Denisarona,JustinWSmith, Unbuttered Parsnip, Happynomad, AlexeyMuranov, Addbot, Aarond144, Jfitzell, NateWessel, Yobot, Jalal0, Ian Kelling,Citation bot, Buenasdiaz, Twri, Kinewma, Miym, Prunesqualer, Mzamora2, JZacharyG, Pmq20, Shadowjams, Hobsonlane, DixonD-Bot, Reaper Eternal, EmausBot, John of Reading, Wikipelli, Bethnim, Mastergreg82, ClueBot NG, EmanueleMinotto, Warumwarum,DavidRideout, BG19bot, Andrey.gric, Szebenisz, ChrisGualtieri, Deltahedron, Jw489kent, Jmerm, Morgoth106, SofjaKovalevskaja andAnonymous: 131

• Graph (mathematics) Source: https://en.wikipedia.org/wiki/Graph_(mathematics)?oldid=668424269 Contributors: The Anome, Man-ning Bartlett, XJaM, Tomo, Stevertigo, Patrick, Michael Hardy, W~enwiki, Zocky, Wshun, Booyabazooka, Karada, Ahoerstemeier, Denfjättrade ankan~enwiki, Jiang, Dcoetzee, Dysprosia, Doradus, Zero0000, McKay, BenRG, Robbot, LuckyWizard, Mountain, Altenmann,Mayooranathan, Gandalf61, MathMartin, Timrollpickering, Bkell, Tobias Bergemann, Tosha, Giftlite, Dbenbenn, Harp, Tom harrison,Chinasaur, Jason Quinn, Matt Crypto, Neilc, Erhudy, Knutux, Yath, Joeblakesley, Tomruen, Peter Kwok, Aknorals, Chmod007, Ab-dull, Corti, PhotoBox, Discospinster, Rich Farmbrough, Andros 1337, Paul August, Zaslav, Gauge, Tompw, Crisófilax, Yitzhak, Kine,Bobo192, Jpiw~enwiki, Mdd, Jumbuck, Zachlipton, Sswn, Liao, Rgclegg, Paleorthid, Super-Magician, Mahanga, Joriki, Mindmatrix,Wesley Moy, Oliphaunt, Brentdax, Jwanders, Tbc2, Cbdorsett, Ch'marr, Davidfstr, Xiong, Marudubshinki, Tslocum, Magister Math-ematicae, Ilya, SixWingedSeraph, Sjö, Rjwilmsi, Salix alba, Bhadani, FlaBot, Nowhither, Mathbot, Gurch, MikeBorkowski~enwiki,Chronist~enwiki, Silversmith, Chobot, Peterl, Siddhant, Borgx, Karlscherer3, Hairy Dude, Gene.arboit, Michael Slone, Gaius Cornelius,Shanel, Gwaihir, Dtrebbien, Dureo, Doetoe, Wknight94, Netrapt, RobertBorgersen, Cjfsyntropy, Burnin1134, SmackBot, Nihonjoe,Stux, McGeddon, BiT, Algont, Ohnoitsjamie, Chris the speller, Bluebot, TimBentley, Theone256, Cornflake pirate, Zven, Anabus, Can'tsleep, clown will eat me, Tamfang, Cybercobra, Jon Awbrey, Kuru, Nat2, Tomhubbard, Dicklyon, Cbuckley, Quaeler, BranStark, Wan-drer2, George100, Ylloh, Vaughan Pratt, Repied, CRGreathouse, Citrus538, Jokes Free4Me, Requestion, Myasuda, Danrah, Robertstead-man, Eric Lengyel, Headbomb, Urdutext, AntiVandalBot, Hannes Eder, JAnDbot, MER-C, Dreamster, Struthious Bandersnatch, JNW,Catgut, David Eppstein, JoergenB, MartinBot, Rettetast, R'n'B, J.delanoy, Hans Dunkelberg, Yecril, Pafcu, Ijdejter, Deor, ABF, Magh-nus, TXiKiBoT, Sdrucker, Someguy1221, PaulTanenbaum, Lambyte, Ilia Kr., Jpeeling, Falcon8765, RaseaC, Insanity Incarnate, Zenek.k,Radagast3, Debamf, Debeolaurus, SieBot, Minder2k, Dawn Bard, Cwkmail, Jon har, SophomoricPedant, Oxymoron83, Henry Delforn(old), Ddxc, Svick, Phegyi81, Anchor Link Bot, ClueBot, Vacio, Nsk92, JuPitEer, Huynl, JP.Martin-Flatin, Xavexgoem, UKoch, Mit-maro, Editor70, Watchduck, Hans Adler, Suchap, Wikidsp, Muro Bot, 3ICE, Aitias, Versus22, Djk3, Kruusamägi, SoxBot III, XLinkBot,Marc van Leeuwen, Libcub, WikiDao, Tangi-tamma, Addbot, Gutin, Athenray, Willking1979, Royerloic, West.andrew.g, Tyw7, Zor-robot, LuK3, Luckas-bot, Yobot, TaBOT-zerem, THEN WHO WAS PHONE?, E mraedarab, Tempodivalse, Пика Пика, Ulric1313,

76 CHAPTER 12. UNORDERED PAIR

RandomAct, Materialscientist, Twri, Dockfish, Anand jeyahar, Miym, Prunesqualer, Andyman100, VictorPorton, JonDePlume, Shadow-jams, A.amitkumar, Kracekumar, Edgars2007, Citation bot 1, DrilBot, Amintora, Pinethicket, CalmerWaters, RobinK, Barras, Tgv8925,DARTH SIDIOUS 2, Powerthirst123, DRAGON BOOSTER, Mymyhoward16, Kerrick Staley, Ajraddatz, Wgunther, Bethnim, Akuta-gawa10, White Trillium, Josve05a, D.Lazard, L Kensington, Maschen, Inka 888, Chewings72, ClueBot NG, Wcherowi, MelbourneStar,Kingmash, O.Koslowski, Joel B. Lewis, Andrewsky00, Timflutre, Helpful Pixie Bot, HMSSolent, Grolmusz, Mrjohncummings, Stevetihi,Канеюку, Void-995, MRG90, Vanischenu, Tman159, Ekren, Lugia2453, Jeff Erickson, CentroBabbage, Nina Cerutti, Chip WildonForster, Yloreander, Manul, JaconaFrere, Monkbot, Hou710, Anon124 and Anonymous: 351

• Multigraph Source: https://en.wikipedia.org/wiki/Multigraph?oldid=663281390 Contributors: Silverfish, McKay, Jerzy, Altenmann,Giftlite, Michael Devore, Robert Weemeyer, Paul August, Kwamikagami, EmilJ, Arthena, Shreevatsa, Bluemoose, SixWingedSeraph,Rjwilmsi, Mike Segal, Nihiltres, Batztown, YurikBot, Gaius Cornelius, Modify, SmackBot, Vaughan Pratt, Myasuda, Sopoforic, ColinRowat, Headbomb, Futurebird, Guy Macon, David Eppstein, Roly Perera, PaulTanenbaum, JP.Martin-Flatin, WestwoodMatt, Farisori,Hans Adler, Hasteur, Dwiddows, Qwfp, Webonfim, Anticipation of a New Lover’s Arrival, The, Addbot, ,ماني Yobot, Danno uk, Citationbot, Twri, Shadowjams, Citation bot 1, Kiefer.Wolfowitz, Redrobsche, 0x24a537r9, Chaotic iak, Accelerometer, MerlIwBot, HelpfulPixie Bot, Alistairwindsor, Lathsim, Eternalko, Nicholasbarry, Jerodlycett and Anonymous: 17

• Operations research Source: https://en.wikipedia.org/wiki/Operations_research?oldid=667016852 Contributors: The Anome, Tback-str, Iwnbap, Khendon, Vignaux, Maury Markowitz, Jdpipe, Michael Hardy, Wshun, Fred Bauder, Dominus, Ixfd64, Zeno Gantner, Ronz,EdH, Hike395, Mydogategodshat, Dysprosia, Jitse Niesen, Penfold, Robbot, Xa4~enwiki, PBS, Gwrede, Jredmond, Altenmann, Hen-rygb, Nilmerg, Aetheling, Jpo, Giftlite, Muness, Oberiko, Mintleaf~enwiki, BenFrantzDale, Fastfission, Leonard G., Gracefool, JustAnother Dan, Andycjp, Piotrus, Togo~enwiki, Joyous!, Fintor, Robin klein, Klemen Kocjancic, Clemwang, Canterbury Tail, Lucidish,Monkeyman, Brianhe, Petrus~enwiki, Obladi~enwiki, RoyBoy, Dungodung, Maurreen, Slambo, Notreadbyhumans, Haham hanuka,Mdd, Msh210, Arthena, Andrewpmk, Bdwilliamscraig, Hu, Czyl, Saxifrage, Novacatz, Eleusis, Myleslong, Pol098, Tabletop, An-maFinotera, Btyner, GraemeLeggett, BDE, Ian Dunster, FlaBot, Mathbot, RobyWayne, Dúnadan, YurikBot, Wavelength, Borgx, An-gus Lepper, RobotE, Encyclops, Arzel, Amckern, Manop, Eddie.willers, Welsh, Gareth Jones, Panscient, Amcfreely, Abrio, CheeseSandwich, Tribaal, Open2universe, Caliprincess, Nelson50, Bluezy, Zvika, That Guy, From That Show!, Sardanaphalus, JJL, SmackBot,DXBari, Elgrandragon, Benjaminevans82, Ohnoitsjamie, Skizzik, Anwar saadat, Sadads, Maxsonbd, Baa, Gracenotes, D nath1, WyckydSceptre, Trekphiler, Tsca.bot, Snowmanradio, Yqwen, Stevenmitchell, RJN, Jon Awbrey, Acdx, Brainfood, Ohconfucius, Vgy7ujm,Aleenf1, Wxm29, Slakr, Beetstra, SandyGeorgia, Ace Frahm, Keith-264, Will Thomas, Vocaro, Philip ea, CRGreathouse, Thomas-meeks, Requestion, Sanspeur, Penbat, Cydebot, Chrislk02, Imajoebob, Jay.Here, Thijs!bot, Surendra mohnot, David from Downun-der, Seaphoto, Matforddavid, Wayiran, Knotwork, Erxnmedia, .anacondabot, Xn4, Swpb, David Eppstein, Jim.henderson, R'n'B, Lit-tleOldMe old, Erkan Yilmaz, NerdyNSK, Afluegel, DadaNeem, Axr15, Jose Gaspar, Alterrabe, Deor, JohnBlackburne, Homarjun,Jimmaths, Toddy1, SueHay, Ask123, BarryList, Rich Janis, PhDinMS, SieBot, BotMultichill, Jyoti buet, Bentogoa, Dralbertomar-quez, Flyer22, Samansouri, Mtrick, Oxymoron83, Wuhwuzdat, S2000magician, Melcombe, Masoudsa, ClueBot, Koczy, Deanlaw, TheThing That Should Not Be, Isaac.holeman, Grantbow, Boing! said Zebedee, EnigmaMcmxc, Three-quarter-ten, PixelBot, Abkesh-vari, JamieS93, TheRedPenOfDoom, Thewellman, Qwfp, Wally Tharg, Graham Sharp, FTGHSmith, Addbot, Fgnievinski, KenK-endall, Fieldday-sunday, Mnh, Leszek Jańczuk, Download, Protonk, LaaknorBot, Lightbot, Zorrobot, Legobot, Luckas-bot, Yobot,VictorK1965, Pcap, AnomieBOT, VanishedUser sdu9aya9fasdsopa, DemocraticLuntz, Bsimmons666, Galoubet, Formol, HanPritcher,TheTechieGeek63, WebsterRiver, Knowledge Incarnate, Lonniev, Isheden, Williamsrus, KosMal, Omnipaedista, Nnhsky, FrescoBot,Krj373, Sidna, D'ohBot, Cargoking, Michael.Forman, Citation bot 1, Shuroo, Kiefer.Wolfowitz, RedBot, Ibizzavic, NorthernCounties, GQian, Duoduoduo, Robinqiu, Earthandmoon, EmausBot, Vader07d, Dramaturgid, JaeDyWolf, Netha Hussain, Erianna, ChuispastonBot,Nrlsouza, 28bot, Snumath89, Will Beback Auto, StopThat, Gareth Griffith-Jones, Mdgarvey, Widr, BradfordF, Helpful Pixie Bot, Merve-unuvar, Wbm1058, BG19bot, Tcody84, Pine, Rjpbi, Compfreak7, Brad7777, Gibbja, MahdiBot, BFL2015, Flower of Mystery, Jbeyerl,Razibot, Randykitty, Kuldeepsheoran1, Pedarkwa, Melody Lavender, Ginsuloft, Lizia7, Pablodim91, Longobardiano, WholeWheatBagel,U2fanboi, Behroozkamali, Bfortz, Sarahfores, Cyberbikerva, Prasoon068, Directorofpubs, KasparBot and Anonymous: 267

• Ordered pair Source: https://en.wikipedia.org/wiki/Ordered_pair?oldid=659428342 Contributors: Damian Yerrick, AxelBoldt, Tarquin,Josh Grosse, Patrick, Chas zzz brown, Wshun, Dominus, Chinju, Minesweeper, Den fjättrade ankan~enwiki, AugPi, Silthor~enwiki, An-dres, Charles Matthews, Dysprosia, Hyacinth, McKay, Aleph4, Robbot, Altenmann, Romanm, MathMartin, Ruakh, Tobias Bergemann,Giftlite, Bogdanb, Wolfkeeper, Ævar Arnfjörð Bjarmason, Gubbubu, Almit39, Discospinster, TedPavlic, Mecanismo, Paul August, Ben-der235, Elwikipedista~enwiki, Chalst, Nickj, Bobo192, Randall Holmes, Smalljim, Obradovic Goran, Sam Korn, JohnyDog, Caesura,Wtmitchell, Oleg Alexandrov, Hoziron, Velho, MattGiuca, Isnow, SixWingedSeraph, Grammarbot, Salix alba, AySz88, Chobot, Bgwhite,YurikBot, Hairy Dude, Michael Slone, Hennobrandsma, Zwobot, Ott2, SmackBot, Mhss, MK8, MalafayaBot, Ladislav Mecir, Tsca.bot,Tamfang, Fagstein, Wvbailey, Tim Q. Wells, Jim.belk, Gabn1, Courcelles, JForget, Sakurambo, CBM, Gregbard, Christian75, Alai-bot, Thijs!bot, AntiVandalBot, .anacondabot, Magioladitis, Soulbot, Usien6, MetsBot, David Eppstein, MartinBot, Anaxial, Iida-yosiaki,LordAnubisBOT, Computer5t, STBotD, VolkovBot, Crisperdue, TXiKiBoT, Martin451, Enviroboy, AlleborgoBot, YonaBot, Radon210,Oxymoron83, Classicalecon, Cbisanidiot, WikipedianMarlith, ClueBot, The Thing That Should Not Be, Bellatrix Kerrigan, Hans Adler,Qwfp, DumZiBoT, Addbot, Pyfan, Betterusername, Wikomidia, Gail, Frmatt, Alfie66, Legobot, Luckas-bot, Yobot, Pcap, Tempodivalse,JRB-Europe, Xqbot, The Fiddly Leprechaun, Isheden, FrescoBot, LucienBOT, Logiker~enwiki, Mfwitten, Negi(afk), Zuiruyu, DrilBot,RandomDSdevel, Pinethicket, Michkol1, MPeterHenry, Aiken drum, Neveln, Energy Dome, Rcsprinter123, Paulmiko, ClueBot NG,This lousy T-shirt, Sn336356, Joel B. Lewis, Nullzero, Jochen Burghardt, Vocaloid frog, I3roly, Muhamad ittal and Anonymous: 108

• Orientation (graph theory) Source: https://en.wikipedia.org/wiki/Orientation_(graph_theory)?oldid=608529513Contributors: Rjwilmsi,David Eppstein, JP.Martin-Flatin and Yobot

• Path (graph theory) Source: https://en.wikipedia.org/wiki/Path_(graph_theory)?oldid=664191818Contributors: Booyabazooka, CharlesMatthews, Dcoetzee, Olego, Dysprosia, Doradus, Zero0000, Altenmann, MathMartin, Tobias Bergemann, Giftlite, MSGJ, SWAdair,Neilc, Vadmium, Gdr, DRE, Chmod007, Rich Farmbrough, Paul August, Derbeth, Oleg Alexandrov, Joriki, Linas, Daira Hopwood, Ad-king80, Jittat~enwiki, YurikBot, Bota47, Macskeeball, Modify, SmackBot, Tsca.bot, ElommolE, Yaninass2, CBM, Flamholz, Thijs!bot,Djr36, Stannered, VictorAnyakin, JAnDbot, Cowrider, David Eppstein, TXiKiBoT, Ashesh0326, Jamelan, Svick, Justin W Smith, Idealgas equation, Addbot, Allliam, Yobot, TaBOT-zerem, Vini 17bot5, Ian Kelling, Citation bot, Twri, Xqbot, Serkan Kenar, Miym, Sas-soBot, Eomund, FrescoBot, Alejandro Erickson, ClueBot NG, KLBot2, Paolo Lipparini, DarafshBot, Mog333, Werddemer, Matroids,Npip99 and Anonymous: 16

• Set (mathematics) Source: https://en.wikipedia.org/wiki/Set_(mathematics)?oldid=668089065 Contributors: Damian Yerrick, Axel-Boldt, Lee Daniel Crocker, Archibald Fitzchesterfield, Uriyan, Bryan Derksen, Zundark, The Anome, -- April, LA2, XJaM, Toby~enwiki,Toby Bartels, Pb~enwiki, Waveguy, LapoLuchini, Youandme, Olivier, Paul Ebermann, Frecklefoot, Patrick, TeunSpaans, Michael Hardy,

12.3. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 77

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78 CHAPTER 12. UNORDERED PAIR

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