Graph (1A) · 19/04/2018 · discrete_mathematics) Complete graph A complete graph is a graph in which each pair of vertices is joined by an edge.

Young Won Lim4/19/18

Graph (1A)

Young Won Lim4/19/18

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Graph (5A) 3 Young Won Lim4/19/18

Some class of graphs (1)

https://en.wikipedia.org/wiki/Graph_(discrete_mathematics)

Complete graphA complete graph is a graph in which each pair of vertices is joined by an edge. A complete graph contains all possible edges.

Connected graphIn an undirected graph, an unordered pair of vertices {x, y} is called connected if a path leads from x to y. Otherwise, the unordered pair is called disconnected.

Bipartite graphA bipartite graph is a graph in which the vertex set can be partitioned into two sets, W and X, so that no two vertices in W share a common edge and no two vertices in X share a common edge. Alternatively, it is a graph with a chromatic number of 2.

Graph (5A) 4 Young Won Lim4/19/18

Complete Graphs

https://en.wikipedia.org/wiki/Complete_graph

Graph (5A) 5 Young Won Lim4/19/18

Connected Graphs

https://en.wikipedia.org/wiki/Connectivity_(graph_theory)

This graph becomes disconnected when the right-most node in the gray area on the left is removed

This graph becomes disconnected when the dashed edge is removed.

With vertex 0 this graph is disconnected, the rest of the graph is connected.

Graph (5A) 6 Young Won Lim4/19/18

Bipartite Graphs

https://en.wikipedia.org/wiki/Bipartite_graph

Example of a bipartite graph without cycles A complete bipartite graph

with m = 5 and n = 3

A graph with an odd cycle transversal of size 2: removing the two blue bottom vertices leaves a bipartite graph.

Graph (5A) 7 Young Won Lim4/19/18

Complete Graphs

https://en.wikipedia.org/wiki/Gallery_of_named_graphs

Graph (5A) 8 Young Won Lim4/19/18

Complete Bipartite Graphs

https://en.wikipedia.org/wiki/Gallery_of_named_graphs

Graph (5A) 9 Young Won Lim4/19/18

Star Graphs

https://en.wikipedia.org/wiki/Gallery_of_named_graphs

Graph (5A) 10 Young Won Lim4/19/18

Wheel Graphs

https://en.wikipedia.org/wiki/Gallery_of_named_graphs

Graph (5A) 11 Young Won Lim4/19/18

Some class of graphs (2)

Planar graphA planar graph is a graph whose vertices and edges can be drawn in a plane such that no two of the edges intersect.

Cycle graphA cycle graph or circular graph of order n ≥ 3 is a graph in which the vertices can be listed in an order v1, v2, …, vn such that the edges are the {vi, vi+1} where i = 1, 2, …, n − 1, plus the edge {vn, v1}. Cycle graphs can be characterized as connected graphs in which the degree of all vertices is 2. If a cycle graph occurs as a subgraph of another graph, it is a cycle or circuit in that graph.

TreeA tree is a connected graph with no cycles.

https://en.wikipedia.org/wiki/Graph_(discrete_mathematics)

Graph (5A) 12 Young Won Lim4/19/18

Planar Graphs

https://en.wikipedia.org/wiki/Planar_graph

A planar graph and its dual

Graph (5A) 13 Young Won Lim4/19/18

Cycle Graphs

https://en.wikipedia.org/wiki/Cycle_graphhttps://en.wikipedia.org/wiki/Gallery_of_named_graphs

Graph (5A) 14 Young Won Lim4/19/18

Tree Graphs

https://en.wikipedia.org/wiki/Cycle_graph

Graph (5A) 15 Young Won Lim4/19/18

Hypercube

https://en.wikipedia.org/wiki/Hypercube

A hypercube can be defined by increasing the numbers of dimensions of a shape:

0 – A point is a hypercube of dimension zero. 1 – If one moves this point one unit length, it will sweep out a line segment, which is a unit hypercube of dimension one. 2 – If one moves this line segment its length in a perpendicular direction from itself; it sweeps out a 2-dimensional square. 3 – If one moves the square one unit length in the direction perpendicular to the plane it lies on, it will generate a 3-dimensional cube. 4 – If one moves the cube one unit length into the fourth dimension, it generates a 4-dimensional unit hypercube (a unit tesseract).

Tesseract

Graph (5A) 16 Young Won Lim4/19/18

Gray Code

https://en.wikipedia.org/wiki/Gray_code

Tesseract

Graph (5A) 17 Young Won Lim4/19/18

Adjacency Lists

https://en.wikipedia.org/wiki/Adjacency_list

Graph (5A) 18 Young Won Lim4/19/18

Incidence Matrix

https://en.wikipedia.org/wiki/Incidence_matrix

Graph (5A) 19 Young Won Lim4/19/18

Adjacency Matrix

https://en.wikipedia.org/wiki/Adjacency_matrix

Graph (5A) 20 Young Won Lim4/19/18

Hamiltonian Path

https://en.wikipedia.org/wiki/Path_(graph_theory)

Graph (5A) 21 Young Won Lim4/19/18

Minimum Spanning Tree

https://en.wikipedia.org/wiki/Minimum_spanning_tree

Graph (5A) 22 Young Won Lim4/19/18

Seven Bridges of Königsberg

https://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg

The problem was to devise a walk through the city that would cross each of those bridges once and only once.

Graph (5A) 23 Young Won Lim4/19/18

Shortest path problem

https://en.wikipedia.org/wiki/Shortest_path_problem

Graph (5A) 24 Young Won Lim4/19/18

Traveling salesman problem

https://en.wikipedia.org/wiki/Travelling_salesman_problem

Graph (5A) 25 Young Won Lim4/19/18

Euler Cycle

Johnsonbough, Discrete Mathematics

Any connected graph with even degree vertices

An Euler cycle

A proof by induction on the number of edges in G

A connected graph Gwith even degree vertices only and k edges (k < n)

A connected graph Gwith even degree vertices onlyand n edges

Assume this is trueAn Euler cycle

An Euler cycleThen this holds true

Graph (5A) 26 Young Won Lim4/19/18

Euler Cycle

Johnsonbough, Discrete Mathematics

Any connected graph with even degree vertices which has n edge

Any connected graph with even degree vertices which has n-1 edge

Any connected graph with even degree vertices which has n-2 edge

Any connected graph with even degree vertices which has 2 edge

Any connected graph with even degree vertices which has 1 edge

An Euler cycle

An Euler cycle

An Euler cycle

An Euler cycle

An Euler cycle

Graph (5A) 27 Young Won Lim4/19/18

Euler Cycle – Base Cases

Johnsonbough, Discrete Mathematics

all even degree vertices

n = 0 edge

n = 1 edge

n = 2 edge an Euler cycle

Graph (5A) 28 Young Won Lim4/19/18

Euler Cycle – decrease the number of edges by one

Johnsonbough, Discrete Mathematics

A connected graph Gwith even degree vertices only and n edges (k < n)

A connected graph G'with even degree vertices only and n-1 edges (k < n)

v1 v2 v3

v1 v2 v3

all even degree vertices

all even degree vertices

P: a path from v to v1

P': a portion of the path P that are in G'

e1 e2

e

Graph (5A) 29 Young Won Lim4/19/18

Euler Cycle – a path from v to v1

v1 v2 v3

v1 v2 v3

v1 v2 v3

v1 v2 v3

v1 v2 v3

v1 v2 v3

P P P

P ' P ' P '

Johnsonbough, Discrete Mathematics

Case 1: P ends at v1 Case 2: P ends at v

2 Case 3: P ends at v

3

v v v

v v v

Graph (5A) 30 Young Won Lim4/19/18

Euler Cycle

en.wikipedia.org

K

J

G

H

F

B

E

D

A

C IABCDEFGHIJK

Graph (5A) 31 Young Won Lim4/19/18

Euler Cycle

2 components

G G'

v1

v2

v3 v1

v2

v3

Graph (5A) 32 Young Won Lim4/19/18

Euler Cycle

1 component

G G'

v1

v2

v3 v1

v2

v3

Graph (5A) 33 Young Won Lim4/19/18

Euler Cycle

2 components

G G'

v1

v2

v3 v1

v2

v3

Graph (5A) 34 Young Won Lim4/19/18

Euler Cycle

1 component

G G'

v1

v2

v3 v1

v2

v3

Young Won Lim4/19/18

References

[1] http://en.wikipedia.org/[2]