1 CHANGING AND UNCHANGING DOMINATION PARAMETERS Dissertation submitted in partial fulfillment of the requirements for the award of the degree of MASTER OF PHILOSOPHY IN MATHEMATICS By SHUNMUGASUNDARI Register Number: 0935313 Research Guide Dr. SHIVASHARANAPPA SIGARKANTI H.O.D., Department of Mathematics Government Science College Nruppathunga Road Bangalore-560 001 HOSUR ROAD BANGALORE-560 029 2010
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1
CHANGING AND UNCHANGING DOMINATION PARAMETERS
Dissertation submitted in partial fulfillment of the requirements for the award of
the degree of
MASTER OF PHILOSOPHY IN MATHEMATICS
By
SHUNMUGASUNDARI
Register Number: 0935313
Research Guide
Dr. SHIVASHARANAPPA SIGARKANTI
H.O.D., Department of Mathematics
Government Science College
Nruppathunga Road
Bangalore-560 001
HOSUR ROAD
BANGALORE-560 029
2010
2
Dr. SHIVASHARANAPPA SIGARKANTI
H.O.D., Department of Mathematics
Government Science College
Kruppathunga Road
Bangalore-560 001
CERTIFICATE
This is to certify that the dissertation submitted by Shunmugasundari on the
title “Changing and Unchanging Domination Parameters” is a record of
research work done by her during the academic year 2009 – 2010 under my
guidance and supervision in partial fulfilment of Master of Philosophy in
Mathematics. This dissertation has not been submitted for the award of any
Degree, Diploma, Associate-ship, Fellowship etc., in this University or in any
other University.
Place: Bangalore Dr. SHIVASHARANAPPA SIGARKANTI
Date: (Guide)
3
DECLARATION
I hereby declare that the dissertation entitled “Changing and Unchanging
Domination Parameters” has been undertaken by me for the award of M.Phil
degree in Mathematics. I have completed this under the guidance of Dr.
SHIVASHARANAPPA SIGARKANTI, H.O.D., Department of Mathematics,
Government Science College, and Kruppathunga Road, Bangalore-560001.
I also declare that this dissertation has not been submitted for the award of
any Degree, Diploma Associate-ship, and Fellowship etc., in this University or in
any other University.
Place: Bangalore Shunmugasundari
Date: (Candidate)
4
ACKNOWLEDGEMENT
Neil Armstrong, the famous Astronaut has said, ‘Research is creating new
Knowledge’. My effort in searching for this knowledge would not have been
complete without the valuable contributions and support of so many benefactors.
I place on my record my gratitude to Dr. (Fr.) THOMAS
C.MATHEW, Vice – Chancellor, Fr. ABRAHAM V.M., Pro Vice-
Chancellor, Prof. Chandrashekaran K.A and Dr. Nanjegowda N. A, Dean of
Sciences for having provided me an opportunity to undertake this research work.
It is with profound gratitude that I acknowledge the constant guidance of
Dr. SHIVASHARANAPPA SIGARKANTI, H.O.D., Department of Mathematics,
Government Science college, Bangalore-560001, Whose valuable guidance,
inspiration, fruitful discussions and constant encouragement at every stage
empowered me to carry out this study and complete this research work
successfully.
I express with all sincerity & regard my deep indebtedness to
Dr. S. PRANESH, Co-ordinator, Post Graduate Department of Mathematics,
Christ University, Bangalore-560 029 for his inspiration, able guidance and
suggestions at every stage of my research work. Without his expertise concern &
benevolent encouragement this work would not have been possible.
I also express my gratitude to Dr. MARUTHAMANIKANDAN S., Post
Graduate Department of Mathematics, Christ University, and Bangalore for his
5
affection and keen interest throughout the course of my work. It is with a sense of
deep appreciation that I place on record my profound thankfulness to him.
I must specially acknowledge Mr. T.V. Joseph, H.O.D., Department of
Mathematics, and other colleagues for their kind co-operation throughout the
period of this study
Finally a special word of thanks to my family members for their
encouragement and support in completing this work.
.
Shunmugasundari
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PREFACE
Graph Theory is a delightful playground for the exploration of proof
techniques in discrete mathematics, and its results have applications in many areas
of computing, social, and natural sciences. How can we lay cable at minimum cost
to make every telephone reachable from every other? What is the fastest route
from the national capital to each state capital? How can n jobs be filled by n
people with maximum total utility? What is the maximum flow per unit time from
source to sink in a network of pipes? How many layers does a computer chip need
so that wires in the same layer don’t cross? How can the season of a sports league
be scheduled into the minimum number of weeks? In what order should a
travelling salesman visit cities to minimum number of weeks? Can we colour the
regions of every map using four colours so that neighbouring regions receive
different colours? These and many other practical problems involve graph theory
(D. B. West, 2002 page1).
Graph Theory was born in 1936 with Euler's paper in which he solved the
Konigsberg Bridge problem. The past 50 years has been a period of intense
activity in graph theory in both pure and applied mathematics. Perhaps the fastest-
growing area within graph theory is the study of domination and related subset
problems, such as independence, covering, and matching.
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This thesis is divided into five chapters, first chapter being the
preliminaries introducing all the terms which are used in developing this thesis. In
this chapter we collect some basic definitions on graphs which are needed for the
subsequent chapters.
In chapter 2 we present a brief review of the historical development of the
study of domination in graphs.
Chapter 3 we deals with dominating set and domination number of a graph.
Some fundamental results on domination are presented. Further several bounds for
the domination number are stated. We also consider a variety of conditions that
might be imposed on a dominating set D in a graph G = (V, E). In this chapter we
will consider a variety of conditions that can be imposed either on the dominated
set V – D, or on V, or on the method by which vertices in V – D are dominated.
In chapter 4 we present the effects on domination parameters when a graph is
modified by deleting a vertex or deleting or adding edges.
In chapter 5 we present many interesting relationships among the six classes
of changing and unchanging graphs.
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Table of Contents S.No Topics Page
No
Preface 1 Preliminaries
1.1 History of Graph theory 1
1.2 Graphs: Basic Definitions 4
1.3 Common Families of Graphs 6
1.4 Isomorphism of Graphs 24
1.5 Trees 25
1.6 Euler Tour and Hamilton cycles 28
1.7 Operation on Graph Theory 31
1.8 Independent set 33
1.9 Matching and Factorization 35
2 Literature survey 38
2.1 Theoretical 44
2.2 New models 45
2.3 Algorithmic 46
2.4 Applications 47
3 Motivation: Theory of Domination in Graphs 55
3.1 Domination number 56
3.2 Independent domination Number 58
3.3 Total domination number 59
3.4 Connected domination number 60
3.5 Connected Total domination number 61
3.6 Clique domination number 62
3.7 Paired domination number 62
3.8 Induced paired domination number 63
3.9 Global domination number 64
3.10 Total global domination number 64
3.11 Edge domination number 65
3.12 Total Edge domination Number 65
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3.13 Connected Edge domination number 66
3.14 Domatic number 67
3.15 Total Domatic Number 67
3.16 Connected Domatic Number 68
3.17 Edge Domatic number 69
3.18 Total Edge Domatic number 71
3.19 Split domination number 72
3.20 Non Split domination number 73
3.21 Cycle non split Dominating Set 74
3.22 Path non split Dominating Set 75
3.23 Cototal Dominating Set 76
3.24 Distance –K Domination 77
4 Changing and unchanging Domination parameters 78
4.1 Terminology 78
4.2 Vertex removal: Changing Domination 84
4.3 Vertex removal: Unchanging Domination 87
4.4 Edge removal :Changing Domination 87
4.4.1 Bondage Number 88
4.4.2 Total Bondage Number 93
4.4.3 Split Bondage Number 97
4.5 Edge Removal: Unchanging Domination 99
4.5.1 Nonbondage Number 100
4.6 Edge addition: Changing Domination 101
4.7 Edge addition: Unchanging Domination 107
4.8 References 108
5 Conclusion 110
5.1 Classes of changing and unchanging graphs 110
5.2 Relationships among Classes 112
6 Bibliography 113
7 Index of symbols 126
8 Index of Definitions 129
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Chapter-1
Preliminaries
In this chapter we collect the basic definitions on graphs which are needed for the
subsequent chapters.
1.1 History of Graph theory
Konigsberg is a city which was the capital of East Prussia but now is known as Kaliningrad in
Russia. The city is built around the River Pregel where it joins another river. An island named
Kniephof is in the middle of where the two rivers join. There are seven bridges that join the
different parts of the city on both sides of the rivers and the island.
People tried to find a way to walk all seven bridges without crossing a bridge twice, but no one
could find a way to do it. The problem came to the attention of a Swiss mathematician named
Leonhard Euler (pronounced "oiler").
11
In 1735, Euler presented the solution to the problem before the Russian
Academy. He explained why crossing all seven bridges without crossing a bridge twice was
impossible. While solving this problem, he developed a new mathematics field called graph
theory, which later served as the basis for another mathematical field called topology
Euler simplified the bridge problem by representing each land mass as a
point and each bridge as a line. He reasoned that anyone standing on land would have to have a
way to get on and off. Thus each land mass would need an even number of bridges. But in
Konigsberg, each land mass had an odd number of bridges. This was why all seven bridges could
not be crossed without crossing one more than once.
The Konigsberg Bridge Problem is the same as the problem of drawing the above
figure without lifting the pen from the paper and without retracing any line and coming back to
the starting point.
Present state of the bridges
Two of the seven original bridges were destroyed by bombs during World War II.
Two others were later demolished and replaced by a modern highway. The three other bridges
remain, although only two of them are from Euler's time (one was rebuilt in 1935). Thus, there
are now five bridges in Konigsberg.
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In terms of graph theory, two of the nodes now have degree 2, and the other two
have degree 3. Therefore, an Eulerian trail is now possible, but since it must begin on one island
and end on the other.
1.2 Graphs: Basic Definitions
In mathematics and computer science, graph theory studies the properties of graphs.
� Mathematical structures used to model pair wise relations between objects from a certain
collection. A "graph" in this context refers to a collection of vertices V (G) or 'nodes' and a
collection of edges E (G) that connect pairs of vertices.
� A graph may be undirected, meaning that there is no distinction between the two vertices
associated with each edge, or its edges may be directed from one vertex to another
Undirected graph:
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Directed graph (Digraph) :
Null graph: Graph that contains no edge is called Null graph because they have null degree of
vertices.
� A null graph of order n is denoted by Nn
Trivial graph: A null graph with only one vertex is called a trivial graph.
� A graph / digraph with only a finite number of vertices as well as finite number of edges
are called a finite graph / digraph; otherwise, it is called an infinite graph / digraph.
� The number of vertices in a (finite) graph is called the order of the graph. It is denoted
by | V | ( The cardinality of the set V)
� The number of edges in a (finite) graph is called the size of the graph. It is denoted by |E |
( The cardinality of the set E)
Loop: If an edge is supported by only one vertex, it is called a loop.
� Two vertices can also have multiple edges.
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� In fact one vertex can have multiple loops.
� The two end vertices are coincident if the edge is a loop
1.3 Common Families of Graphs:
Simple Graph: A graph with no loops or multiple edges is called a simple graph.
Multigraph: A graph which contains multiple edges but no loops is called a multigraph.
General graph: A graph which contains multiple edges or loops (or both) is called a general
graph.
Pseudo graph: A multi graph in which loops are allowed is called a pseudo graph.
� Every edge has two end vertices; every edge is incident on two vertices.
� We also say that Vertex A incident with edge e and Vertex B incident with edge e.
Degree of vertex (Valency): Let G is the graph with loops, and let v be a vertex of G. The
degree of v is the number of edges meeting at v, and is denoted by deg (v).
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Deg (A) = 3 Deg (B) = 4 Deg (H) = 1
� An isolated vertex has zero degree.
� Let G be a multi graph. The maximum degree of G, denoted by ∆∆∆∆(G), is denoted as the
maximum number among all vertex degrees in G.
∆∆∆∆(G) = max {d (v) /v εεεε V (G)}
� Let G be a multi graph. The minimum degree of G, denoted by δδδδ(G), is denoted as the
minimum number among all vertex degrees in G.
δδδδ(G) = min {d (v) /v εεεε V (G)}
Ex.
Here ∆∆∆∆(G) = 4 and δδδδ(G) = 1
Regular graph: A graph G is said to be regular if every vertex in G has the same degree
� G is said to be k-regular if d(v) = k for each vertex v in G, Where k ≥≥≥≥ 0.
� An edge is incident only on two vertices.
� A vertex may be incident with any number of edges.
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� Two non-parallel edges are said to be adjacent edges if they are incident on a common
vertex.
� Two vertices are said to be adjacent vertices (or neighbors) if there is an edge joining
them.
� The set of all neighbors of v in G is denoted by N(v);
i.e. N (v) = {x | x is a neighbor of v}.
N (A) = {B, C, D} and N (B) = {D, E}
Petersen graph: The Petersen graph is the 3-regular graph. It posses a number of graph theoretic
properties and it frequently used to illustrate established theorems and to test conjectures.
� A graph G is k-regular if and only if ∆(G) = δ(G) = k
Walk: A walk in a multigraph G is an alternating sequence of vertices and edges beginning and
ending at vertices:
v0 e0v1e1 v2 e2v3e3 . . . vk-1 ek-1vk,
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Where k ≥ 1 and ei is incident with vi and vi+1, for each i = 0,1,2 , . . . , k-1.
� The walk is also called a v0 – vk walk with its initial vertex v0 and terminal vertex vk.
� The length of the walk is defined as ‘k’, which is the number of occurrences of edges in
the sequence.
Trail: A walk is called a trail if no edge in it is traversed more than once.
Path: A walk is called a path if no vertex in it is visited more than once.
� walk (1) is neither a trail nor a path
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� walk (2) is a trail but not a path
� walk (3) both a trail and a path
� Every path must be a Trail.
� A u-v walk is said to be closed if u = v, that is, its initial and terminal vertices are the
same; and open otherwise.
� A closed walk of length at least two in which no edge is repeated is called a circuit.
Connected graph: A multi graph G is said to be connected if every two vertices in G are joined
by a path. Otherwise it is disconnected.
� Every disconnected graph can be split up into a number of connected sub graphs, called
components.
Ex: Connected non simple graph
Ex : Disconnected non simple graph
� Let G be a connected multi graph, and u, v be any two vertices in G. The distance from
u to v , denoted by d(u , v) is the smallest length of all u-v paths in G ( This is also
known as geodesic distance)
� The greatest distance between any two vertices in a graph G
(i.e.) max {d (u, v) / u, v Є V (G)}
is called the diameter of G and it is denoted by diam (G).
19
� The eccentricity ε of a vertex v is the greatest distance between v and any other vertex.
� The radius of a graph is the minimum eccentricity of any vertex.
Density: The density of G is the ratio of edges in G to the maximum possible number of edges
Density = 2L/ (p * (p-1))
Where L is the number of edges in the graph and p is the number of vertices in the graph.
Density = 2*7 / (7*6) = 1 / 3
Bouquet: A Graph consisting of a single vertex with n self loops is called a bouquet and is
denoted Bn.
Ex. B4
Dipole: A Graph consisting of two vertices and n edges joining them is called a dipole and is
denoted Dn
Ex. D5
The Complete Graphs: A simple graph of order ≥ 2 in which there is an edge between every
pair of vertices is called a complete graph (or a full graph)
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In other words, a complete graph is a simple graph in which every pair of distinct vertices is
adjacent. It is denoted by Kn.
Ex. K5
Path Graph : A path graph P is a simple connected graph with |Vp| = |Ep | + 1 that can be drawn
so that all of its vertices and edges lie on a single straight line and it is denoted by Pn .
Ex. P8
Circular ladder graph: The Circular ladder graph CLn is visualized as two concentric n-cycles
in which each of the n pairs of corresponding vertices is joined by an edge.
Ex. CL4
Cut point: A vertex is a cut point if its removal increases the number of components in the
graph.
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Bridges: An edge is a bridge if its removal increases the number of components in the graph.
Vertex-connectivity: The connectivity κ (G) of a connected graph G is the minimum number of
vertices that need to be removed to disconnect the graph (or make it empty).
κ (G) = 1
Edge-connectivity: The edge-connectivity λ (G) of a connected graph G is the minimum
number of edges that need to be removed to disconnect the graph.
λ(G) = 2
Block: A block of a loop less graph is a maximal connected subgraph H such that no vertex of H
is a cut vertex of H.
Ex. G:
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G has four blocks; they are the subgraph induced on the vertex subsets {u, v, w, x}, {x, y}, {y, z,
m}.
Block graph: The block graph of a graph G, denoted by BL (G), is the graph whose vertices
correspond to the blocks of G, such that two vertices of BL (G) are joined by a single edge
whenever the corresponding blocks have a vertex in common.
Ex. G: BL (G):
Bipartite graph (or bigraph): A bipartite graph is a graph whose vertices can be divided into
two disjoint U and V such that every edge connects a vertex in U to one in V; that is, U and V
are independent sets.
Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles.
Complete bipartite graph:
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Complete bipartite graph or biclique is a special kind of bipartite where every vertex of
the first set is connected to every vertex of the second set. The complete bipartite graph with
partitions of size
| V1 | = m and | V2 | = n, is denoted Km,n.
Star: A star Sk is the complete bipartite graph K1, k.
Ex. K1, 7
Wheel: The wheel graph Wn is a graph on n vertices constructed by connecting a single vertex to
every vertex in an (n-1)-cycle.
Ex. W8
Planar graph: A planar graph is that can be embedded in the plane, i.e., it can be drawn on the
plane in such a way that its edges intersect only at their endpoints.
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The Cycles: A graph of order n ≥ 3 is called cycle if its n vertices can be named as v1 ,v2 ,…,vn
such that v1 is adjacent to v2 , v2 is adjacent to v3 to vn-1 is adjacent to vn , vn is adjacent to v1 ,
and no other adjacency exists; that is ,
V (G) = { v1, v2 ,…, vn } and
E (G) = {v1v2 , v2v3, , … ,vnv1 }
A cycle of order n is denoted Cn, we call Cn an n-cycle.
Ex.C6
Girth: The minimum length of a cycle in a graph G is the girth g (G).
g (G) = 3
25
Unicyclic Graph: A connected graph containing exactly one cycle
Subgraphs: A subgraph S of a graph G is a graph such that
� The vertices of S are a subset of the vertices of G.
(i.e.) V(S) ⊆ V (G)
� The edges of S are a subset of the edges of G.
(i.e.) E(S) ⊆ E (G)
� S is a subgraph of G
� S1 is not a subgraph of G
Proper subgraph: If S is a subgraph of G then we write S ⊆ G. When S ⊆ G but S ≠ G.
i.e. V(S) ≠ V(G) or E(S) ≠ E(G), then S is called a Proper subgraph of G .
A spanning subgraph: A spanning subgraph of G is a subgraph that contains all the vertices of
G.
( i.e.) V(S) = V(G)
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S is spanning subgraph of G
A vertex induced Subgraph: A vertex-induced subgraph is one that consists of some of the
vertices of the original graph and all of the edges that connect them in the original denoted by ⟨
V⟩.
� G1 is an induced subgraph - induced by the set of vertices
V1 = {A,B,C,F} .
� G2 is not an induced subgraph.
An edge-induced subgraph: An edge-induced subgraph consists of some of the edges of the
original graph and the vertices that are at their endpoints.
Some graph operation:
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Vertex deleted subgraph: For any vertex v of graph G, G-v is obtained from G by removing v
and all the edges of G which have v as an end. G-v is referred to as a vertex deleted subgraph.
Edge deleted subgraph: If G = (V,E) and e is an edge of G then G-e is obtained from G by
removing the edge e (but not its end point(s) . G-e is referred to as a edge deleted subgraph.
Complement of a graph: The complement of the graph G, denoted by , is the graph with V
( ) = V(G) such that two vertices are adjacent in if and only if they are not adjacent in
G. ( interchanging the edges and the non-edges)
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Clique: clique in an undirected graph G is a subset of the vertex set C ⊆ V, such that for every
two vertices in C, there exists an edge connecting the two. This is equivalent to saying that the
subgraph induced by C is complete (in some cases, the term clique may also refer to the
subgraph).
A maximal clique is a clique that cannot be extended by including one more adjacent vertex,
that is, a clique which does not exist exclusively within the vertex set of a larger clique.
A maximum clique is a clique of the largest possible size in a given graph. The clique number
ω (G) of a graph G is the number of vertices in the largest clique in G.
ω (G) = 5
1.4 Isomorphism of Graphs:
The simple graphs G1 = (V1, E1) and G2 = (V2, E2) are isomorphic if there is a bijection (an one-
to-one and onto function) f from V1 to V2 with the property that a and b are adjacent in G1 if and
only if f (a) and f(b) are adjacent in G2, for all a and b in V1.Such a function f is called an
isomorphism.
In other words, G1 and G2 are isomorphic if their vertices can be ordered in such a way that the
adjacency matrices MG1 and MG2 are identical
� For two simple graphs, each with n vertices, there are n! possible isomorphism
� For this purpose we can check invariants, that is, properties that two isomorphic simple
graphs must both have
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• the same number of vertices,
• the same number of edges, and
• The same degrees of vertices.
Note that two graphs that differ in any of these invariants are not isomorphic, but two graphs that
match in all of them are not necessarily isomorphic
Example : Are the following two graphs isomorphic?
Solution:
Yes, they are isomorphic, f(a) = e, f(b) = a, f(c) = b, f(d) = c, f(e) =d.
� If G is isomorphic to H, then V(G ) = V(H) and E(G) = E(H).
Adjacency matrix: Let G = (V, E) be a simple graph with |V| = n. Suppose that the vertices of
G are listed in arbitrary order as v1, v2… vn. The adjacency matrix A (or AG) of G, with respect
to this listing of the vertices, is the n×n zero-one matrix with 1 as it’s (i, j) entry when vi and vj
are adjacent, and 0 otherwise.
In other words, for an adjacency matrix A = [aij],
aij = 1 if {vi, vj} is an edge of G,
aij = 0 otherwise.
Example: What is the adjacency matrix AG for the following graph G based on the order of
vertices a, b, c, d?
30
� Adjacency matrices of undirected graphs are always symmetric.
Incidence matrix: Let G = (V, E) be an undirected graph with |V| = n. Suppose that the vertices
and edges of G are listed in arbitrary order as v1, v2… vn and e1, e2… em, respectively. The
incidence matrix of G with respect to this listing of the vertices and edges is the n×m zero-one
matrix with 1 as it’s (i, j) entry when edge ej is incident with vi, and 0 otherwise.
In other words, for an incidence matrix M = [mij],
mij = 1 if edge ej is incident with vi
mij = 0 otherwise.
Example: What is the incidence matrix M for the following graph G based on the order of
vertices a, b, c, d and edges 1, 2, 3, 4, 5, 6?
� Incidence matrices of directed graphs contain two 1s per column for edges connecting
two vertices and one 1 per column for loops.
1.5 Tree: A tree is a graph in which any two vertices are connected by exactly one simple path.
In other words, any connected graph without cycles is a tree.
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Spanning tree: A spanning tree T of a connected , undirected graph G is a tree composed of all
the vertices and some (or perhaps all) of the edges of G.
Informally, a spanning tree of G is a selection of edges of G that form a tree spanning every
vertex. That is, every vertex lies in the tree, but no cycles (or loops) are formed
Forest: A forest is an undirected graph, all of whose connected components are trees; in other
words, the graph consists of a disjoint union of trees. Equivalently, a forest is an undirected
cycle-free graph.
Galaxy: A galaxy is a forest in which each component is a star.
1.6 Euler Tour and Hamilton cycles:
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Euler path: A graph is said to be containing an Euler path if it can be traced in 1 sweep without
lifting the pencil from the paper and without tracing the same edge more than once. Vertices may
be passed through more than once. The starting and ending points need not be the same.
Euler circuit: An Euler circuit is similar to an Euler path, except that the starting and ending
points must be the same.
� The term Eulerian graph has two common meanings in graph theory. One
meaning is a graph with an Eulerian circuit, and the other is a graph with every vertex of
even degree
� An Eulerian path, Eulerian trail or Euler walk in an undirected graph is a path
that uses each edge exactly once. If such a path exists, the graph is called traversable or
semi-eulerian.
� An Eulerian cycle, Eulerian circuit or Euler tour in an undirected graph is a
cycle that uses each edge exactly once. If such a cycle exists, the graph is called
unicursal. While such graphs are Eulerian graphs, not every Eulerian graph possesses an
Eulerian cycle.
Let's look at the graphs below; do they contain an Euler circuit or an Euler path?
33
What is the relationship between the nature of the vertices and the kind of path/circuit that the
graph contains? We will have the answer after looking at the table below.
Graph Number of odd
vertices
Number of even
vertices
What does the
path contain?
(Euler path = P;
Euler circuit = C;
Neither = N)
1 0 10 C
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2 0 6 C
3 2 6 P
4 2 4 P
5 4 1 N
6 8 0 N
From the above table, we can observe that:
� A graph with all vertices being even contains an Euler circuit.
� A graph with 2 odd vertices and some even vertices contains an Euler path.
� A graph with more than 2 odd vertices does not contain any Euler path or circuit.
Hamiltonian path: Hamiltonian path (or traceable path) is a path in an undirected graph which
visits each vertex exactly once.
Hamiltonian cycle: A Hamiltonian cycle (or Hamiltonian circuit) is a cycle in an undirected
graph which visits each vertex exactly once and also returns to the starting vertex
Hamiltonian graph: A graph is Hamiltonian if it contains a Hamilton cycle.
1.7 Some Operations on Graph Theory:
Union: There are several ways to combine two graphs to get a third one. Suppose we have
graphs G1 and G2 and suppose that G1 has vertex set V1 and edge set E1, and that G2 has vertex
35
set V2 and edge set E2. The union of the two graphs, written G1 U G 2 will have vertex set V1 U
V2 and edge set E1U E2.
If we choose the null graph N1 and the complete graph K5 we will get the graph in following
figure
N1 U K5
Sum (Join): The sum of two graphs G1 and G2, written G1 + G2, is obtained by first forming the
union G1UG2 and then making every vertex of G1 adjacent to every vertex of G2.
N1 + K5
Graph Cartesian Product: The Cartesian graph product G = G1 X G2, sometimes simply called
"the graph product” of graphs G1 and G2 with disjoint point sets V1 and V2 and edge sets E 1
and E 2 is the graph with point set and adjacent with whenever
or
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1.8 Independent set: An independent set or stable set is a set of vertices in a graph no two of
which are adjacent.
{I, D}, {I, D, F} and {H, C, E} are some of the independent sets.
But {A, D, F} and {A, C, H} are not. Independent sets are also called disjoint or mutually
exclusive.
Maximum independent set: A maximum independent set is a largest independent set for a
given graph G.
Maximal independent set:
A maximal independent set or maximal stable set is an independent set that is not a
subset of any other independent set.
Ex. In the cycle C10
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The sets {B,F,I} ,{A,C,E,G,I} , {A,C,E} are some of the independent sets.
{J,C,F,H} ,{A,C,E,G,I} ,{B,D,F,H,J} are maximal independent set.
{J, C, F, H} is not a maximum independent set.
Independence number ββββ0 (G): The number of vertices in a maximum independent set of G is
called the independence number of G and is denoted by ββββ0 (G).
ββββ0 (G) = 4
Independent set of edges: An independent set of edges of G has no two of its edges are
adjacent
Ex. K4
Edge independence number ββββ1 (G): The number of edges in a maximum independent
set of G is called the edge independence number of G and is denoted by β1 (G).
β1 (G) = 2
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Point cover: A vertex and a line are said to cover each other if they are incident. A set of points
which covers all the lines of graph G is called a point cover.
Vertex covering number: The smallest number of points in any vertex cover for G is called its
vertex covering number and it is denoted by α0 (G).
Edge Cover: A set of lines which covers all the vertices of graph G is called a line cover.
Edge covering number: The smallest number of lines in any edge cover for G is called its
edge covering number and it is denoted by α1 (G).
α0 (G) = 3 and α1 (G) = 3
1.9 Matching: Given a graph G, a matching M in G is a set of pair wise non-adjacent edges; that
is, no two edges share a common vertex.
A vertex is matched (or saturated) if it is incident to an edge in the matching. Otherwise the
vertex is unmatched (or unsaturated). A maximal matching is a matching M of a graph G
with the property that if any edge not in M is added to M, it is no longer a matching, that is, M is
maximal if it is not a proper subset of any other matching in graph G.
A maximum matching is a matching that contains the largest possible number of edges. There
may be many maximum matching. The matching number ν (G) of a graph G is the size of a
maximum matching. Note that every maximum matching is maximal, but not every maximal
matching is a maximum matching..
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ν (G) = 2
A perfect matching is a matching which matches all vertices of the graph.
A near-perfect matching is one in which exactly one vertex is unmatched. This can only occur
when the graph has an odd number of vertices, and such a matching must be maximum.
���� An alternating path is a path in which the edges belong alternatively to the matching (M)
and not to the matching (E-M).
���� An augmenting path is an alternating path that starts from and ends on free (unmatched)
vertices.
Factorization: A factor of a graph G is a spanning subgraph of G which is not totally
disconnected. G is the sum of factors G i if it is their line disjoint union, and such a union is
called a factorization of G.
���� An n-factor is a regular of degree n.
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���� If G is the sum of n-factors, their union is called an n-factorization and G itself is n-
factorable.
���� A 1-factorization of a graph is a decomposition of all the edges of the graph into 1-
factors.
G: K4
G = G1 + G2 + G3
���� A 2-factor is a collection of cycles that spans all vertices of the graph.
G: K5
G1: G2:
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G = G1 + G2
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References:
1. Bollobas.B,Graph Theory: An Introductory Course Springer 1979.
λ(G) Edge-connectivity, 16 d(u , v) Geodesic distance, 12 g (G) Girth, 19 γ g (G) Global domination number, 64
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M Incidence matrix, 26
k0 (G) Independence number , 34 γ i (G) Independent domination number, 58 γ ip (G) Induced paired domination number, 63 < D> Induced subgraph, 21 ν (G) Matching number, 36 b - s (G) Negative split bondage number, 98 N(v) Neighbor, 9 γ ns (G) Non split domination number, 73
γ p (G) Paired domination number, 62
Pn Path graph γ pns (G) Path non split domination number, 75 G-e Removal of a edge,23 G-v Removal of a point, 22 b s (G) Split bondage number, 97 γ s (G) Split domination number, 72 Sk Star, 18
γ ss (G) Strong split domination number, 73
γ sns (G) Strong non split domination number, 74 S Subgraph, 22 |E | The cardinality of the set e, 5 | V | The cardinality of the set v, 5 E (G) The edge set of g, 4 ∆(G) The maximum degree of g, 7 δ(G) The minimum degree of g, 7 b n (G) The nonbondage number, 99 Nn The null graph of order n, 5 V (G) The vertex set of g, 4 b t(G) Total bondage number, 93
dt(G) Total domatic number, 46 γt (G) Total dominating number, 59
γ t 1(G) Total edge domination number, 65
γ tg(G) Total global domination number, 64 α0 (G) Vertex covering number, 35 κ (G) Vertex-connectivity, 15