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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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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
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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").
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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).
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� 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
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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?
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� 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?
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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
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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
38
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..
39
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.
40
���� 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:
41
G = G1 + G2
42
References:
1. Bollobas.B,Graph Theory: An Introductory Course Springer 1979.
2. C. Berge Graphs, North-Holland 1985.
3. Chartrand.G, Introductory Graph Theory, Dover 1985 .
4. Diestel.R, Graph Theory, Springer-Verlag 1997.
5. F. Harary, Graph Theory, Addison Wesley, Reading, MA, (1969).
6. G. Chartrand and L.Leniak, Graphs Digraphs, Fourth Edition, CRC Press, Boca Raton,
2004.
7. Gould.RJ Benjamin/Cummings, Graph Theory , 1988
8. Gross. JL and Yellen. J, Graph Theory and its Applications, CRC Press LLC, 1998.
9. J. A. Bondy and U. S. R. Murthy, Graph Theory with Applications, Macmillan, London.
10. J. Clark and D. A. Holton, a first look at graph theory, World Scientific Pub. Singapore /
Allied Pub.Ltd. New Delhi (1995)
11. J. Wilson and J.J. Watkins John, Graphs: An Introductory Approach, Wiley & Sons 1990.
12. M. Behzad, A characterization of total graphs, Proc. Amer. Math. Soc., 26 (1970) 383 –
389.
13. M. Capobianco and J.C. Molluzzo, Examples and Counterexamples in Graph Theory,
North-Holland 1978.
14. O. Ore, Theory of Graphs, AMS Colloquium Publications 38 AMS 1962.
15. R. C. Brigham and D. Dutton, On Neighborhood graphs, J. Combinatorics Inf & Syst.
Sci., 12 (1987) 75 – 85.
16. R. L. Brooks On coloring the nodes of a network, Proc.Cambridge Philos. Soc. 37
(1941) 194 – 197.
43
17. R.J. Wilson Introduction to Graph Theory by R.J. Trudeau Dover Publications, 1994.
18. S. Arumugam and S. Ramachandran, Invitation to Graph Theory Scitech Publications
(2001).
19. T. Gallai, Uber extreme Punkt-und Kantenmenger, Ann Univ. Sci. Budapest, Eotvos
Sect. Math, 2 (1959) 133 – 138.
20. V. R. Kulli, Graph Theory, Vishwa Internat. Publications, (2000).
21. West.DB, Introduction to Graph Theory, Prentice Hall 1996.
44
45
Chapter-3
Concept of Domination in graphs
In this chapter we collect the basic definitions and theorems on domination in graphs which are
needed for the subsequent chapters.
3.1 Dominating set : In graph theory, a dominating set for a graph G = (V, E) is a subset D of
V such that every vertex not in D (every vertex in V- D ) is joined to at least one member of D
by some edge.
(i.e.) A set D of vertices in a graph G is called a dominating set of G if every
vertex in V-D is adjacent to some vertex in D.
Ex. In the following graph G
The set D = {A, B, E, H} is one of the dominating set
Minimum Dominating set:
A dominating set D is said to be Minimum Dominating set if D consist of minimum
number of vertices among all dominating sets..
Ex. In the following graph G
46
Domination number:
The domination number γ (G) is the number of vertices in a smallest dominating set for G.
(The cardinality of minimum dominating set)
Ex. In the following graph G
47
Minimal Dominating Set:
A dominating set D is called Minimal dominating set if no proper subset of D is a
dominating set
Ex.
The sets {B,C,E} ,{D,C} and {B,E,F,G} are Minimal dominating sets.
In the following graph
The set D1 = {B, C, D} is a dominating set. But D1 is not a minimal dominating set.
D2 = {C, D} is a minimal dominating set. Also D2 is a minimum dominating set.
A minimum dominating set is a minimal dominating set, but the converse is not always true.
Theorem 2.1: A dominating set D is a minimal dominating set if and only if for each vertex
v∈D, one of the following two conditions holds:
(a) v is an isolated vertex of D
(b) there exists a vertex u ∈ V-D such that N(u) ∩ D = {v}.
Theorem 2.2: Every connected graph G of order n ≥ 2 has a dominating set D whose
complement V-D is also a dominating set.
48
Theorem 2.3: If G is a graph with no isolated vertices, then the complement V-D of every
minimal dominating set D is a dominating set.
2.2 Independent dominating set:
A dominating set D of a graph G is an independent dominating set if
the induced sub graph <D> has no edges.
Ex.
Independent domination Number: γ i (G):
The independent domination number γi (G) of a graph G is the minimum
cardinality of an independent dominating set.
γ i (G) = 2
Theorem 2.4: An independent set is maximal independent set if and only it is independent and
dominating.
Theorem 2.5: Every maximal independent set in a graph G is a minimal dominating set.
Theorem 2.6[: For any graph G,
p /(1 +∆(G) ≤ γ (G) ≤ p - ∆(G).
where p is the number of vertices in V (G).
2.3 Total dominating set:
49
A dominating set D of a graph G is a total dominating set if the
induced sub graph <D> has no isolated vertices. i.e. Every vertex of G is adjacent to at
least one vertex in D
Ex. P7
Total domination number γ t (G):
The total domination number is the minimum cardinality of a total dominating set
Here γt (G) = 4
Theorem 2.7: If G is a connected graph with p ≥ 3 vertices then γt (G) ≤ 2p/3.
Theorem 2.8: If G has p vertices and no isolates, then γt (G) ≤ p - ∆(G) +1.
Theorem 2.9: If G is connected and ∆(G) < p-1, then γt (G) ≤ p - ∆(G) .
2.4 Connected Dominating set:
A dominating set D is said to be connected dominating set if induced subgraph <D> is
connected.
Connected domination number γ c (G):
The connected domination number is the minimum cardinality of a connected
dominating set.
Ex. P8
50
γ c (P8) = 6
Theorem 2.10: If G is a connected graph with p ≥ 3 vertices then γc (G) ≤ p-2
Theorem 2.11: For any connected graph G, p/(∆(G)+1) ≤ γc (G) ≤ 2q-p. Furthermore, the lower
bound is attained if and only if ∆(G) = p-1 and the upper bound is attained if and only if G is a
path.
Theorem 2.12: For any connected graph G, γc (G) ≤ p-∆(G).
2.5 Connected Total dominating set:
A total dominating set D of a graph G is a connected total dominating set if
the induced sub graph <D> is connected.
Connected Total domination number γ ct (G):
The connected total domination number γ ct (G) is the minimum cardinality of a
connected total dominating set.
γ ct (G) = 4
Theorem 2.13: For any connected graph G with p ≥ 4,
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p/(∆(G)+1) + 1≤ γct (G) ≤ 2p-q. Furthermore, the lower bound is attained if G = Kp and
the upper bound is attained if and only if G is a path.
Theorem 2.14: For any connected graph G with p ≥ 4, γct (G) ≤ p-2.
Theorem 2.15: If T is a tree of order p ≥ 4 and T ≠ K 1,p-1, then γct (G) = p – e. where e is the
number of end vertices of a tree.
2.6 Clique dominating set:
A dominating set D of a graph G is a dominating clique if the induced sub
graph <D> is a complete graph.
Clique domination number γ cl (G):
The Clique domination number γ cl is the minimum cardinality of a dominating
clique.
γ cl (G) = 4
2.7 Paired dominating set:
A dominating set D of a graph G is a paired dominating set if the induced
sub graph ⟨D⟩ contains at least one perfect match.
Paired domination number γ p (G):
The paired domination number γ p (G) is the minimum cardinality of a paired
dominating set.
52
G:
γ p (G) = 4
Theorem 2.16: If G has no isolated vertices, then 2 ≤ γ p (G) ≤ p and these bounds are sharp.
Theorem 2.17: If G has no isolated vertices, then p/∆(G) ≤ γ p (G).
Theorem 2.18: If a connected graph G has p ≥ 6 and δ(G) ≥ 2, then γ p (G) ≤ 2 p/3
2.8 Induced Paired dominating set:
A dominating set D of vertices of a graph G is an induced paired dominating set
if the induced sub graph ⟨D⟩ is a set of independent edges.
Induced paired domination number γ ip (G):
The induced paired domination number γ i p (G) is the minimum
cardinality of an induced paired dominating set of G.
Ex. G:
γ ip (G) = 1
2.9 Global dominating set:
A dominating set D of a graph G is a global dominating set if D is also a
dominating set of
Global domination number γ g (G):
53
The global domination number γ g (G) is the minimum cardinality of a global
dominating set.
γ g (G) = 2
2.10 Total Global dominating set:
A total dominating set D of a graph G is a total global dominating set if
D is also a total dominating set of .
Total global domination number γ tg(G):
The total global domination number γ tg(G) is the minimum cardinality of a total
global dominating set.
γ tg(G) = 4
Theorem 2.19: Let G be a graph such that neither G nor have an isolated vertex. Then
2q-p(p-3) ≤ γ tg(G)
Theorem 2.20: Let G be a graph such that neither G nor have an isolated vertex. Then
γ tg(G) ≤≤≤≤ 2α0(G).
2.11 Edge dominating set :
54
A set F of edges in a graph G is an edge dominating set, if every edge not in F (every
edge in E-F) is adjacent to at least one edge in F.
Edge domination number γ 1(G):
The edge domination number γ 1(G) of a graph G is the minimum cardinality of an edge
dominating set of G.
γ 1(G) = 3
2.12 Total Edge dominating Set:
A set F of edges in a graph G is called a total edge dominating set of G if for every edge
in E is adjacent to at least one edge in F.
i.e. a set F of edges in G is called total edge dominating set of G if for every edge e∈
E, there exists an edge e1 ∈ F such that e and e1 have a vertex in common.
Total Edge domination Number: γ t 1(G)
The minimum cardinality of a total edge dominating set of G is the total edge
domination number of a graph G, and it is denoted by γ t 1(G).
γ t 1 (G) = 3
2.13 Connected Edge Dominating set :
55
A edge dominating set D is said to be connected edge dominating set if induced
subgraph <F> is connected.
Ex. P8
Connected Edge domination number γ c 1(G):
The connected edge domination number is the minimum cardinality of a
connected edge dominating set.
γ c 1(G) = 5
2.14 Domatic number: d(G)
The domatic number is defined as the maximum number of disjoint dominating set.
Ex.
X = { {B,E}, {A,D,G},{C,F}} and d(G) = 3
Theorem 2.21: For any graph G, d (G) ≤ δ(G) +1.
Theorem 2.22: For any graph G, d (G) = 1 if and only if G has an isolated vertex.
Theorem 2.23: For any graph G, p +(p - δ(G)) ≤ d(G).
2.15 Total Domatic Number:
56
A partition ∆ of a vertex V of G is called total domatic partition of G if
each class of ∆ is a total dominating set of G. The maximum number of classes of total domatic
partition of G is called the total domatic number G and is denoted by dt(G).
Ex. G:
X = {{A, B}, {D, C}}
dt (G) = 2
Theorem 2.24: For any graph G without isolated vertices, dt (G) ≤ d (G).
Theorem 2.25: For any graph G without isolated vertices, dt (G) ≤ δ(G).
Theorem 2.26: If K p is a complete graph with p ≥ 2 vertices then dt (K p ) = p/2
Theorem 2.27: For any graph G without isolated vertices, d(G) /2 ≤ dt (G).
2.16 Connected Domatic Number:
A partition ∆ of a vertex V of a connected graph G is called a connected domatic
partition of G if each class of ∆ is a connected dominating set of G. The maximum number of
classes of connected domatic partition of G is called the connected domatic number G and is
denoted by dc(G).
Ex. G:
57
X = {{A, B}, {H,I} , {D,E} }
d c(G) = 3
Theorem 2.28: For any connected graph G, dc (G) ≤ δ(G).
Theorem 2.29: For any connected graph G which is not compete dc (G) ≤ δ(G)+1.
Theorem 2.30: For any connected graph G which is not compete dc (G) ≤ κ(G).
2.17 Edge domatic number:
An edge domatic partition of G is a partition of E(G), all of whose classes are edge
dominating sets in G. The maximum number of classes of an edge partition of G is called the
edge domatic number of G and is denoted by d1(G).
Ex. C6
X = {{e1, e4 } , {e2 , e5 }, {e3 , e6 }}
d1 (G) = 3
Theorem 2.31: If Pp is a path with p ≥ 3 vertices then d1 (Pp) = 2.
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Theorem 2.32: If Cp is a cycle with p ≥ 3 vertices then d1 (Cp) = 3 if p is divisible 3
=2 otherwise.
Connected Edge domatic number:
A connected edge domatic partition of G is a partition of E (G), all of whose classes are
connected edge dominating sets in G. The maximum number of classes of a connected edge
partition of G is called the connected edge domatic number of G and is denoted by dc1 (G).
Ex. G:
X = {{e2, e3, e4}, {e7, e11, e9, e10}}
dc1 (G) = 2.
2.18 Total Edge domatic number:
A total edge domatic partition of G is a partition of E(G), all of whose classes are total
edge dominating sets in G. The maximum number of classes of a total edge partition of G is
called the total edge domatic number of G and is denoted by dc1(G).
Ex. G: W6
59
X = { {e2 , e8, e11 , e5 } , {e3 , e9, e12 , e6 } , {e4 , e10, e7 , e1 }}
dt1 (G) = 4.
2.19 Split Dominating Set:
A dominating set D of G is a split dominating set if the induced subgraph <V-D> is
disconnected.
Ex. C5
Split domination number γ s (G):
The split domination number is the minimum cardinality of a split dominating set.
γ s (G) = 2
Theorem 2.33: For any graph G, γ s (G) ≤ α0 (G).
Theorem 2.34: For any graph G, γ (G) + γ s (G) ≤ p.
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Theorem 2.35: γ s (Cp) = p/3 if p ≥ 4;
γ s (W p) = 3 if p ≥ 5;
γ s (K m ,n) = m if 2 ≤ m ≤ n.
Strong split Dominating Set:
A dominating set D of G is a strong split dominating set if the induced sub
graph <V-D> is totally disconnected with at least two vertices.
Strong split domination number γ ss (G): The strong split domination number is the minimum
cardinality of a strong split dominating set
Ex. G:
γ ss (G) = 4
2.20 Non Split Dominating Set:
A dominating set D of G is a non split dominating set if the induced sub
graph <V-D> is connected.
Non Split domination number γ ns (G):
The split domination number is the minimum cardinality of a non split
dominating set.
γ ns (G) = 5
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Theorem 2.36: If T is a tree which is not a star then γ ns (T) ≤ p-2.
Theorem 2.37: If T is a tree with p ≥ 3 then p – m ≤ γ ns (T). Here m is the number of vertices
adjacent to end vertices.
Strong non split Dominating Set:
A dominating set D of G is a strong split dominating set if the induced
subgraph <V-D> is complete.
Strong non Split domination number γ sns (G):
The strong non split domination number is the minimum cardinality of a
strong non split dominating set.
γ sns (G) = 3
2.21 Cycle non split Dominating Set:
A dominating set D of a connected graph G is a cycle non split
dominating set if the induced sub graph <V-D> is cycle in G.
Cycle non split domination number γ cns (G):
The cycle non split domination number is the minimum cardinality of a
cycle non split dominating set
62
γ cns (G) = 2
Preposition 2.38: For any connected graph G with p ≥ 4, γ (G) ≤ γ cns (G).
Preposition 2.39: For any connected graph G with p ≥ 4, γ (G) + γ cns (G) ≤ p.
2.22 Path non split Dominating Set:
A dominating set D of a connected graph G is a path non split
dominating set if the induced sub graph <V-D> is a path in G
Path non split domination number γ pns (G):
The path non split domination number is the minimum cardinality of a path
non split dominating set.
Ex. C5
γ pns (G) = 3
Preposition 2.40: For any nontrivial connected graph G, γ (G) ≤ γ pns (G).
Preposition 2.41: For any nontrivial connected graph G, γ (G) + γ pns (G) ≤ p.
63
Preposition 2.42: γ pns (K p) = p – 2, p ≥ 3.
γ pns (Cp) = p – 2, p ≥ 3.
γ pns (Pp) = p – 2, p ≥ 3.
γ pns (W p) = 2, p ≥ 4.
γ pns (Km ,n ) = m + n – 3 , m ≥ 2, n ≥ 3.
2.23 Cototal Dominating Set:
A dominating set D of G is a cototal dominating set if the induced
subgraph <V-D> is has no isolated vertices.
Cototal domination number γ cot (G):
The cototal domination number γ cot (G) is the minimum cardinality
cotal dominating set.
γ cot (G) = 3
Theorem 2.43: For any graph G, p – (2/3)q ≤ γ cot (G).
Theorem 2.44: let G be a graph such that each component of G is not a star. Then
γ cot (G) ≤ p - δ(G).
Theorem 2.45: For any graph G, 2(p –q) – p0 ≤ γ cot (G). Where p0 is the number of isolated
vertices in G.
2.24 Distance –K Domination:
Given any integer k ≥ 1, vertex subset D is a distance-k dominating set of a
graph G if for all v Є VG –D, there exists x Є D such that d(v , x) ≤ k
64
Ex. A Minimum distance -2 dominating set
The distance-k domination number:
The distance-k domination number of a graph G, denoted dk –dom(G) , is the cardinality
of a minimum distance-k dominating set of G. More over dk –dom(G) ≤ ϒ(G)
ϒ(G) = 4
d2–dom(G) = 2
65
References:
1. A. M. Barcalkin and L. F. German, The external stability number of the Cartesian product
of graphs, Bul. Akad. Stiince RSS Moldoven, No 1, 94 (1979) 5-8.
2. B. Bollobas and E.J.Cockayne, Graph theoretic parameters concerning domination,
independence and irredundance, J. Graph Theory 3 (1979) 241-250.
3. C. Berge, Theory of Graphs and its Applications, Methuen, London, (1962).
4. E. J. Cockayne, S.T.Hedetniemei and D.J.Miller, Properties of hereditary hyper graphs and
middle graphs, Canad. Math. Bull., 21 (1978) 461-468.
5. E. Sampathkumar and H. B. Walikar, The connected domination number of a
graph, J. Math.Phys.Sci., 13 (1979) 607-613.
6. F. Harary and M. Livingston, Characterization of trees with equal domination and
independent domination number. Congr. Number, 55 (1986) 121-150.
7. H. B. Walikar, B. D. Acharya and E. Sampathkumar, Recent developments in the theory of
domination in graphs. In MRI Lecture Notes in Math. Metha Research Inst., Allahabad
No. 1, (1979).
8. K. Seyffarth and G. Macgillivray, Domination numbers of planar graphs, J. Graph
Theory, 22 (1996) 2134-229.
9. L. A. Sanchis, Maximum number of edges in connected graph with given domination
number, Discrete Math. 87 (1991) 65-72.
10. O. Favaron, A bound on the independent domination number of a tree. Vishwa Internat. J.
Graph Theory, 1 (1992) 19-27.
11. R. B. Allan and R. C. Laskar, On domination and independent domination numbers of a
graph, Discrete Math, 23 (1978) 73-76.
12. S. L. Mitchell and S. T. Hedetniemi, Edge domination in trees. Congr. Numer.19 (1997)
489-509.
13. S. R. Jayaram, Edge domination in graphs, a Graphs Combin. 3(1987) 357-363.
66
14. T.W. Haynes and P. J. Slater, Paired Domination in graphs, Networks, 32 (1998) 199-206.
15. V. G. Vizing, Some unsolved problems in graph theory, Uspekhi Mat. Nauk. 23 (6(144))
(1968) 117-134.
16. V. R. Kulli, Theory of domination in graph, Vishwa Internat.Publications,2010.
67
Chapter-3
Literature survey
The following problem can be said to be the origin of the study of dominating sets in
graphs. The following figures illustrates a standard 8 x 8 chessboard on which is placed a queen.
According to the rules of chess a queen can, in one move, advance any number of squares
horizontally, vertically, or diagonally (assuming no other chess piece lies in its way). Thus, the
queen in the above figure can move to (or attack, or dominate) all of the squares marked with an
‘X’. In the 1850s, chess enthusiastics in Europe considered the problem of determining the
minimum number of queens that can be placed on a chess board so that all squares are either
attacked by a queen or are occupied by a queen. The following figure illustrates a set of six
68
queens which together attack or dominate, every square on the board. It was correctly thought in
the 1850s, that five is the minimum number of queens that can dominate all of the squares of an
8 x 8 chessboard.
Case-1 : No two Queens attack each other
69
Case-2 : All Queens lie on the main diagonal
Case-3 : All Queens lie on a common column
70
Mathematical history of Domination in Graphs:
The mathematical study of dominating sets in graphs began around 1960.
The subject has historical roots dating back to 1862, when Jaenisch studied the problem of
determining the Minimum number of Queens which are necessary to cover an n x n chess
board.
Among others, the following are the 3 basic types of problems:
Covering: What is the minimum number of chess pieces of given type which is necessary to
cover / attack /dominate every square of an n x n board?
This is an example of the problem, finding a dominating set of minimum cardinality.
Independent Covering: What is the minimum number of mutually non attacking chess pieces
of a given type which are necessary to dominate every square of an n x n board?
This is an example of the problem of finding a minimum cardinality independent
dominating set.
Independence: What is the maximum number of chess pieces of a given type which can be
placed on an n x n chessboard in such a way that no two of them attack / dominate each other?
This is an example of the problem of finding the maximum cardinality of an
independent set.
These three problem types were studied in detail by Yaglom and Yaglom brothers around
1964
� In 1958 Claude Berge wrote a book on graph theory, in which he defined for the
first time, the concept of the domination number of a graph ( he called this number as ‘
the coefficient of external stability ‘)
� In 1962, Oystein Ore published his book on Graph theory in which he used for
the first time, the names ‘ dominating set ‘ and ‘ dominating number ‘ .
71
� In 1977, Cockayne and Hedetniemi publish a survey of the few results known at
that time, about dominating sets and graph. In this survey, they were the first to use the
notation γ (G) for the domination number in a graph, which subsequently became the
accepted notation.
� This survey paper seems of have set in motion the modern study of domination
in graphs
� Some twenty years later, more than 1200 research papers have been published on
this topic and the number of papers is steadily growing.
According to S. T. Hedetniemi, R. C. Laskar, they divide the contributions in Topics on
domination theory into three sections, entitled ‘theoretical’, ‘new models’ and ‘algorithmic’.
� The nine theoretical papers retain a primary focus on properties of the standard
domination number ϒ(G)
� The four papers which they classify as ‘ new models ‘ are concerned primarily
with new variations in the domination theme.
� The eight algorithmic papers are primarily concerned with finding classes of
graphs for which the domination number, and
� Several other domination-related parameters can be computed in polynomial
time.
3.1 Theoretical:
For a variety of reasons they lead of this volume with the paper “ Chessboard
domination problem “ by Cockayne, because Cockayne has done the most definitive work in
this area. The follow up paper “ On the queen domination problem” by Ginstead, Hahne and
Vanstone the best approximation to the old problem of placing a minimum number of queens
on an arbitrary nxn chessboard so that all squares are ‘covered’ by atleast one queen.
72
. It is able to present next a reprint of a paper by Berge and Duchet entitled “Recent
problems and results about Kernels in directed graph” Claude Berge has done more than
anyone in particular of domination theory. He used the terminology “Coefficient of external
stability” instead “The domination number”
. David Sumner was one of the early researcher in domination theory and was perhaps
the first one to consider the question of domination in critical graphs. In this paper “ Critical
concepts in domonation” he considers the problem of characterizing graphs for which adding
any edge e decreases the domination number. He also considers the problem of characterizing
graphs having minimum dominating sets D which are independent. i.e. no two vertices in D are
adjacent.
A related notion, By Fink , Jacobson, Kinch and Roberts in “The bondage number
of graph”, is that of finding a set of edges F of smallest order (called the bondage number),
whose removal increases the domination number.
In the original survey paper on domination Cockayne and Hedetniemi
introduced the domatic number of a graph denoted d(G) which equals the maximum order of a
partition{V1 ,V2, V3,…, VR } of V(G) such that every set Vi is a dominating set. Today Zelinka
has become the world’s foremost authority on the domatic number and a related partition
numbers. He has published nearly two dozen papers on this topic. Zelinka entitled “ Regular
totally domatically full graphs” and Rall entitled “ Domatically critical and domatically full
graphs “. On the domatic number of a graph.
3.2 New models:
The concepts of domination, covering and centrality are so interrelated. In a 1985
paper, Hedetniemi, Hedetniemi and Laskar list 30 different types of domination. As of now
twice as many types of domination problem have been studied.
The paper “ Dominating cliques in graphs ” by Cozzens and Kelleher, studies
the existence of families of graphs which contain a complete subgraph whose vertices form a
dominating set. They present several forbidden subgraph conditions which are sufficient to imply
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the existence of dominating cliques and they present a polynomial algorithm for finding a
domination clique for a certain class of graphs.
The paper “Covering all cliques of a graph” by Tuza considers a different kind of
domination, in which one seeks a minimum set of vertices which dominates all cliques(i.e.
maximal complete subgraphs ) of a graph.
The paper by Brigham and Dutton entitled “ Factor domination in graphs”
considers, the general problem of finding a minimum set of vertices which is a dominating set of
every subgraph in a set of edge disjoint subgraphs, say G1 ,G2, G3,…, Gt , whose union is a given
graph G.
The paper by Sampathkumar entitled “The least point covering and domination
number of a graph” is one of many papers in which one imposes additional conditions on a
dominating set, e.g. the dominating set must induce a connected subgraph(connected
domination), a complete subgraph (dominating clique), or a totally-disconnected graph
(independent domination). In Sampathkumar’s paper the domination number of the subgraph
induced by the dominating set must be minimized.
3.3 Algorithmic:
Nearly 100 papers containing domination algorithm or complexity results have been
published in the last 10 years. Perhaps, the first domination algorithm was an attempt by Daykin
and Ng in 1966 to compute the domination number of an arbitrary tree. But their algorithm
seems to have an error that cannot be easily corrected.
Cockayne, Goodman and Hedetniemi apparently constructed the first domination
algorithm for trees in 1975 and, at about the same time, David Johnson constructed the first
(unpublished) proof that the domination problem for arbitrary graphs is NP complete.
The first paper by Corneil and Stewart entitled “ Dominating sets in perfect
graphs” presents both a brief survey of algorithmic results on domination and a discussion of
the dynamic-programming-style technique that is commonly used in designing domination
algorithms, especially as they are applied to the family of perfect graph.
74
The paper “Unit disk graphs” by Clark , Colbourn and Johnson discusses the
algorithmic compelxity of such problem as domination , independent domination and connected
domination , and several other problems, on the intersection graphs of equal size circles in the
plane. This paper is significant since it contains the result that the Domination problem for grid
graphs, a subclass of unit disk graphs, is NP-complete.
The paper “ Permutation graphs: Connected domination and Steiner trees” by Colbourn
and Stewart , a variety of NP-complete problems have been shown to have polynomial solutions
when restricted to permutation graph.
The paper “ The discipline number of a graph” Chavatal and Cook, provides an
example of the relatively recent study of fractional( i.e. real valued ) parameters of graphs. These
are the values obtained by real relaxations of the integer linear programs corresponding to
various graphical parameters like domination, matching, covering and independence.
The paper “Best location of service centers in a tree- like network under budget
constraints” by McHugh and Perl, provides both a nice applications perspective on domination
and an illustration of the many papers that have been published on the topic of centrality in
graphs. It also provides an example of a pseudo-polynomial domination algorithm and another
example of the dynamic programming technique applied to domination problems.
The paper “Dominating cycles in Halin graphs” by Skowronska and
Syslo, discusses both a fourth class of graphs on which polynomial time domination algorithms
can be constructed, and the notion of a dominating cucle, i.e. a cycle C in a graph such that
every vertex not in C lies at most one from some vertex in C.
The paper “Finding dominating cliques efficiently, in strongly chrodal
graphs and undirected path graph” by Kratsch is an algorithmic mate of the paper by
Cozzens and Kelleher on dominating cliques, find the dominating cliques of minimum size.
The paper “On minimum dominating sets with minimum intersection” by
Grinstead and Slatter, which is a good representative of the fast developing area of polynomial,
75
and even linear , algorithms on partial K-trees. Grinstead and Slatter introduce a difficult, new
type of problem, prove that it is in general NP-complete, and give a linear time solution when
restricted to trees. This solution also uses a dynamic programming style approach and a
methodology created by Wimer in his 1987 Ph.D. Thesis.
3.4 Applications:
School Bus Routing:
Most school in the country provide school buses for transporting children to and from school.
Most also operate under certain rules, one of which usually states that no child shall have to walk
farthrer than, say one quarter km to a bus pickup point. Thus, they must construct a route for
each bus that gets
� Within one quarter km of every child in its assigned area.
� No bus ride can take more than some specified number of minutes, and
� Limits on the number of children that a bus can carry at any one time.
Let us say that the following figure represents a street map of part of a city, where each edge
represents one pick up block. The school is located at the large vertex. Let us assume that the
school has decided that no child shall have to walk more than two blocks in order to be picked up
by a school bus. Construct a route for a school bus that leaves the school, gets within two blocks
of every child and returns to the school. One such simple route is indicated by the directed edges
in the following figure
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A second possible route is indicated below. With this route the school bus can
turn around and drive back down a street. Both routes define what are called distance-2
dominating sets in the sense that every vertex not on the route(not in the set) is within distance
two (two edges) of at least one point on the route. These routes also define what are called
connected dominating sets in the sense that the set of shaded vertices on the route forms a
connected subgraph of the entire graph. The connected domination number ϒc (G) equals the
minimum cardinality of a dominating set D such that <D> is connected
77
Computer Communication Networks:
Consider a computer network modelled by a graph G = (V,E), for which vertices
represents computers and edges represent direct links between pairs of computers. Let the
vertices in following figure represent an aray, or network, of 16 computers, or processors. Each
processors to which it is directly connected. Assume that from time to time we need to collect
information from all processors. We do this by having each processor route its information to
one of a small set of collecting processors (a dominating set). Since this must be done relatively
fast, we cannot route this information over too long a path. Thus we identify a small set of
processors which are close to all other processors. Let us say that we will tolerate at most a two
unit delay between the time a processors sends its information and the time it arrives at a nearby
collector. In this case we seek a distance-2 dominating set among the set of all processors. The
two shaded vertices form a distance-2 dominating set in the hypercube network in following
figure
78
Radio Stations:
Suppose that we have a collection of small villeges in a remote part of the world. We would
like to locate radio stations in some of these villeges so that messages can be broadcast to all of
the villages in the region. Since each radio station has a limited broadcasting range, we must use
several stations to reach all villages. But since radio stations are costly, we want to locate as few
as possible which can reach all other villages.
Let each village be represented by a vertex. An edge between two villages is labelled with the
distance, say in kilometers, between the two villages
79
Let us assume that a radio station has a broadcast range of fifty kilometers. What is the least
number of stations in a set which dominates (within distance 50) all other vertices in ths graph?
A set (B,F,H,J} of cardinality four is indicated in the following figure(b).
Here we have assumed that a radio station has a broadcast range of only fifty kilometers, we can
essentially remove all edges in the graph, which represent a distance of more than fifty
kilometers. We need only to find a dominating set in this graph.Notice that if we could afford
80
radio stations which have a broadcast range of seventy kilometers, three radio stations would
sufficient.
Other Applications:
� Locate TV and Mobile communication towers
� Wireless Ad Hoc Network
� Mining Scientific Data Sets Using Graphs
� Assignment problem
� Guard location problem
� Surveillance system and Coding theory
� Land Surveying.
References:
1. T. W. Haynes, S. T. Hedetniemi, and P. J. Slater (eds), Fundamentals of Domination
in Graphs, Marcel Dekker, Inc. New York, 1998.
2. T. W. Haynes, S. T. Hedetniemi, and P. J. Slater (eds), Domination in Graphs:
Advanced Topics, Marcel Dekker, Inc. New York, 1 998.
3. S.T.Hedetniemi, and R. C. Laskar, Topics on domination, Annals of discrete
mathematics.
81
82
Chapter-4
Changing and unchanging Domination parameters
An Important Consideration in the topological design of a network is fault tolerance,
that is, the ability of the network to provide service even when it contains a faulty component or
components. The behavior of a network in the presence of a fault can be analyzed by determining the
effect that removing an edge (link failure) or a vertex (processor failure) from its underlying graph G
has on the fault- tolerance criterion. For example, a ϒ-set in G represents a minimum set of
processors that can communicate directly with all other processors in the system. If it is essential for
file servers to have this property and that the number of processors designated as file servers be
limited, then the domination number of G is the fault-tolerance criterion. In this example, It is most
important that ϒ (G) does not increase when G is modified by removing a vertex or an edge. From
another perspective, networks can be made fault-tolerant by providing redundant communication
links (adding edges). Hence, we examine the effects on ϒ (G) when G is modified by deleting a
vertex or deleting or adding an edge
4.1 Terminology:
The semi-expository paper by Carrington, Harary, and Haynes surveyed the
problems of characterizing the graphs G in the following six classes. Let G-v (respectively, G-e)
denote the graph formed by removing vertex v (respectively, edge e) from G. We use acronyms
to denote the following classes of graphs (C represents changing; U represents unchanging;
V: vertex; E: Edge; R: removal; A: addition).
(CVR) γ (G - v) ≠ γ(G) for all v ∈ V
(CER) γ (G - e) ≠ γ(G) for all e ∈ E
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(CEA) γ (G + e) ≠ γ(G) for all e ∈ E(G )
(UVR) γ (G - v) = γ(G) for all v ∈ V
(UER) γ (G - e) = γ(G) for all e ∈ E
(UEA) γ (G + e) = γ(G) for all e ∈ E(G )
These six problems have been approached individually in the literature with other terminology.
Hence we examine them and several related problems using the above “changing and
unchanging” terminology first suggested by Harary [F. Harary, Changing and unchanging
invariants for graphs. Bull. Malaysian Math. Soc. 5 (1982) 73-78.
It is useful to partition the vertices of G into three sets according to how their
removal affects their γ (G). Let V = V 0 ∪ V
+ ∪ V -
for
V 0 = {v ∈ V : γ (G - v) = γ (G)}
V + = {v ∈ V : γ (G - v) > γ (G)}
V - = {v ∈ V : γ (G - v) < γ (G)}
Similarly, the edge set can be partitioned into
E 0
= {uv ∈ E: γ (G -uv) = γ (G)}
E + = {uv ∈ E: γ (G -uv) > γ (G)}
For Example, the graphs in the following figure G:
84
γ (G) = 2
The graph G1 : G-{X}
γ (G-{X}) = 2
V 0 = {X ∈ V: γ (G – {X} = γ (G)}
(i.e.) V 0 = {A, B, C, D, E, X} ----------------------------------→1
The graph G2: G-{F}
85
γ (G-{F}) = 6
γ (G-{F}) > γ(G)
V + = {F ∈ V : γ (G – {F} > γ (G)}
(i.e.) V + = {F} ------------------------------------� 2
The graph G3: G-{H}
γ (G-{H}) = 1
γ (G-{H}) < γ(G)
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V - = { H ∈ V : γ (G – {H} < γ (G)}
i.e. V - = {F} -----------------------------------------------------�3
From equation 1, 2 and 3
V = V 0 ∪ V
+ ∪ V
-
Vertex removal:
� The removal of vertex v from a graph G results a graph G-v such that
γ (G-v) > γ(G)
γ (G-v) < γ(G)
γ (G-v) = γ(G)
� The removal of vertex from G can increase γ(G) by more than one
Ex. For the graph K1,4
Here γ (G) = 1 γ (G-A) = 4
γ(G-A) > γ(G)
� But the removal of vertex from G can decrease γ(G) by at most one
Ex. For the cyclic graph C4
87
Here γ(G) = 2 γ(G- v) = 1
γ(G-v) < γ(G)
Edge removal:
� The removal of an edge from a graph G can increase by the domination number by at
most one and cannot decrease the domination number. ( i.e.) γ(G-e) = γ(G) + 1
Ex. G:
γ(G) = 3 γ(G-e) = 4
γ(G-e) = γ(G) + 1
� The domination number is unchanged when any single edge is removed.
γ(G-e) = γ(G)
Ex. For the cyclic graph C8
88
Here γ(G) = 3 γ(G- e) = 3
γ(G-e) = γ(G)
Vertex removal: Unchanging Domination
Ex. For the cyclic graph C8
Here γ (G) = 3 γ(G- v) = 3
γ (G-v) = γ(G)
4.2 Vertex removal: Changing Domination:
The vertices in V + were characterized by Bauer, Harary, Nieminen and Suffel [1]
Theorem 4.1[1]: A vertex v ∈ V + if and only if
(i) . v is not an isolated vertex and is in every γ-set of G, and
(ii). no subset S⊆ V – N (v) with cardinality γ(G) dominates G-v.
89
The vertices in V - were characterized by Sampathkumar and Neeralagi.
Theorem 4.2 [2]: A vertex v ∈ V - if and only if pn [v, D] = {v} for some γ-set D containing v.
Carrington et al. determined the properties of V +
and V – and showed that for any graph G in changing
vertex removal, γ (G - v) < γ (G) for all v ∈ V, that is , V = V – and V
+ = Φ.
Theorem 4.3 [3]: For any graph G,
(a) If v ∈ V + , then for every γ-set D of G, v ∈ D and pn[v,D] contains at least two non adjacent
vertices,
(b) if x ∈ V + and
y ∈ V
- , then x and y are not adjacent,
(c) |V0 | ≥ 2|V
+ |,
(d) γ (G) ≠ γ (G - v) for all v ∈ V if and only if V = V – , and
(e) if v ∈ V – and v is not an isolated vertex in G , then there exists a γ-set D of G such that v not in D
Brigham, Chinn and Dutton determined a sufficient condition to imply that γ (G - v) = γ(G).They
established the following theorem.
Theorem 4.4 [4]: If a graph G has a non isolated vertex v such that the subgraph induced by N (v) is
complete, then γ (G - v) = γ (G).
Theorem 4.5[22]: If a graph G∈CVR and γ (G) ≥ 2, then diam (G) ≤ 2(γ (G) – 1).
Bauer et al. [1] studied a problem of determining the minimum number of vertices whose
removal changes γ(G). Let µµµµ+ denote the minimum number of vertices whose removal increases
the domination number and µµµµ - denote the minimum number of vertices whose removal
decreases the domination number They obtained the following results.
Theorem 4.6[1]: For any tree T, µµµµ+ (T) = 2 if and only if there are vertices u and v such that
(1) every γ-set contains either u or v.
(2) v is in every γ-set for T-u and u is in every γ-set for T-v.
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They also established the following results
Theorem 4.7[1]: For any graph
(a) µ -(G) ≤ γ(G) + 1.
(b) min { µ+(G), µ
-(G)} ≤ δ(G) + 1.
(c) If G has an end vertex, then µ+(G ≥ 2 implies µ
-(G) ≤ 2.
(d) For n ≥ 7, µ +( Pn ) + µ
-( Pn) = 4.
(e) For n ≥ 8, µ +( Cn ) + µ
-( Cn ) = 6.
Bauer, Harary, Nieminen and Suffel showed that V 0
is never empty for a tree. They proved the
following theorem.
Theorem4. 8[1]: For any tree T with n ≥ 2, there exists a vertex v ∈∈∈∈ V, such that γ (T –v} = γ (T).
4.3 Vertex removal: Unchanging Domination
Carrington, Harary and Haynes [3] characterized graph for which γ (G –v} = γ (G).
Theorem 4.9[3]: A graph G has γ (G –v} = γ (G) for any vertex v ∈ V if and only if G has no isolated
vertices and for each vertex v, either
(i) There is a γ-set D1such that v∉ D
1 and for each γ-set D such that v ∈ D, pn[v,D] contains at
least one vertex from V-D, or
(ii) v is in every γ-set and there is a subset of γ (G) vertices in G-N[v] that dominating G-v
4.4 Edge removal (CER- Edge removal: Changing Domination)
The removal of an edge from G cannot decrease γ-set and can increase it by at most one. Thus a
graph for which the domination number changes when an arbitrary edge is removed has the
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property that γ (G –e} = γ (G) + 1, for all e∈E. The graphs in CER were called “γ+- critical graphs”
and independently characterized by Bauer et al. [1] and Walikar and Acharya [5].
Theorem 4.10 [1, 5]: A graph G has γ (G –e} = γ (G) + 1 for any edge e∈E if and only if each
component of G is a star.
4.4.1 Bondage Number: In a communications network, network consists of existing communication
links between a fixed set of sites. The problem at hand is to select a smallest set of sites at which to place
transmitters is joined by a direct communication link to one that does have a transmitter. This problem
reduces to finding a minimum dominating set in the graph, corresponding to this network, that has a
vertex corresponding to each site, and an edge between two vertices if and only if the corresponding sites
have a direct communications link joining them.
Suppose that someone does not know which sites in the network act as transmitters, but does
know that the set of such sites corresponds to a minimum dominating set in the related graph. What is the
fewest number of communication links that he must sever so that at least one additional transmitter would
be required in order that communication with all sites be possible? With this in mind, they introduce the
bondage number of a graph
This concept was introduced by Fink et al. [6] with the above application in mind.
Bondage number: b(G)
The bondage number b(G) of a non empty graph G is the minimum cardinality among all
sets of edges X ⊆ E such that γ(G-X) > γ(G).
Ex. K4
γ(G) = 1 γ(G-X) =2
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γ (G-X) > γ(G)
b (G) = 2
First results on the bondage number are found in Bauer at al. [1] (the term “edges stability
number” was used instead of bondage number).
Hartnel and Rall [7] characterized the trees having bondage number 2.
Theorem 4.11[1, 5]: If T is a nontrivial tree then b(T) ≤ 2.
Theorem 4.12 [1, 5]: If any vertex of a tree T is adjacent with two or more end vertices then
b (T) = 1.
Theorem 4.13: If F is a forest, then F is an induced subgraph of a tree S with b(S) = 1 and a tree
T with b(T) = 2.
Fink et al. [6] gave the exact values for b (G) of selected graphs.
Theorem 4.14[6]: The bondage number of the complete graph K p (p ≥2) is
b (K p) = p/2 .
Ex. K 5
γ(K 5 ) = 1 γ(K 5 - x ) = 2
γ (K 5 - x ) > γ(K 5 )
b (K p) =3.
Theorem 4.15[6]: The bondage number of the p-cycle C p is
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b (C p) = 3 if p ≡ 1 (mod 3),
2 otherwise
Ex. C 6
We know that γ(C p ) = p/3 for p ≥ 3
γ (C6 - x ) = 3 and γ(C 6 ) = 2
γ (C6 - x ) > γ(C 6 )
b (C6) =2.
Theorem 4.16[6]: The bondage number of the path of order p (p ≥2) is
b (P p) = 2 if p ≡ 1 (mod 3),
1 otherwise
Ex. P 7
We know that γ (P p ) = p/3 for p ≥ 1
γ (P7 - x ) = 4 and γ(P 7 ) = 3
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γ(P7 - x ) > γ(P 7 )
b (P7) =2.
General bounds: In this section we shall establish bounds on the bondage number of a graph
that are independent of the graph’s structure,
Theorem 4.17: If G is connected graph of graph of order p ≥ 2 then b(G) ≤ p-1.
Theorem 4.18: If G is a nonempty graph, then
b (G) ≤ min { deg u + deg v – 1: u and v are adjacent vertices}.
The above theorem is proved by Bauer, Harary, Nieminen and Suffel.
Corollary of the above theorem: If ∆(G) and δ(G) denote respectively the maximum and
minimum degree among all vertices of nonempty connected graph G, then
b (G) ≤ ∆(G) + δ(G) - 1
Another bound on the bondage number that involves the maximum degree among the vertices of
the graph is given by the following theorem. This bound also indicates a relationship between the
bondage number and the domination number.
Theorem 4.18[6]: If G is nonempty graph with domination number γ(G) ≥ 2, then
b (G) ≤ (γ(G) – 1) ∆(G) + 1.
Theorem 4.19[6]: If G is a connected graph of order p ≥ 2, then
b (G) ≤ p - γ(G) + 1.
The following upper bound was established by Hartnell and Rall [8].
Theorem 4.20 [6]: If G has edge connectivity λ(G) ≥ 1, then
b (G) ≤ ∆(G) + λ(G) - 1.
Conjecture: It is conjectured in [6] that b (G) ≤ ∆(G) + 1 for any nonempty graph G.
95
In 2000, Kang and Yuan [20] made a breakthrough toward this conjecture. They obtained
the following results.
Theorem 4.21[20]: If G is a connected planar graph then b (G) ≤ max {8, ∆(G) +2}.
Theorem 4.22[20]:If G is a connected planar graph with no degree five vertex, then b(G)≤ 7
Theorem 4.23 [7] If G is a connected planar graph then
b(G) ≤ 6, if g(G) ≥ 4;
5, if g(G) ≥ 5;
4, if g(G) ≥ 6;
3, if g(G) ≥ 8.
Theorem 4.24[8]: Hartnell and Rall [8] proved that b(Gn) = 3/2 * ∆(Gn) for the Cartesian
product Gn = Kn × Kn, which disproves the existence of an upper bound of the form
b (G) ≤∆(G) + c for any constant number c.
Theorem 4.25[18]: Teschner [18] showed that b(G) ≤3/2 * ∆(G) holds for any graph G
satisfying ϒ(G) ≥ 3.
Theorem 4.26[19]: S. Klavzar et al. determined in [19] the domination number of Cm × Cn for
some m, n, they obtained that ϒ(C3× Cn) = n –(n/4) for n ≥ 4.
Theorem 4.27: For any positive integer k, we have
b(C3 × C4k) = 2;
b(C3 × C4k+1) = 4; and
b(C3× C4k+2) = 4.
Theorem 4.28: For any positive integer k, we have
b(C3 × C4k+3) = 5 for every k ≥ 1.
4.4.2 Total Bondage Number: b t(G)
96
The total bondage number b t (G) of a graph G is the minimum cardinality among all
sets of edges X ⊂ E such that γ t(G-X) > γ t(G).
Ex. P7
γ t (G) = 4
γ t(G-X) = 5
γ t(G-X) > γ t(G) b t(G) = 1
This concept was introduced by Kulli and Patwari [12]. They determined b t(G) for some
families of graph.
Theorem 4.29[12]: The total bondage number of the complete graph K p (p ≥5) is
b t (K p) = 2p - 5.
Ex. K 6
We know that γ t(K p ) = 2 for p ≥ 2
γ t(K6 - x ) = 3 and γ t (K6 ) = 2
γ t(K6 - x ) > γ t (K6 )
97
b t (K 6) =7.
Theorem 4.30[12]: The total bondage number of the p-cycle C p (p ≥3) is
b t (C p) = 3 if p ≡ 2 (mod 4),
2 otherwise.
Theorem 4.31[12]: The total bondage number of the path of order (p ≥4) is
b t(P p) = 2 if p ≡ 2 (mod 4),
1 otherwise.
Theorem 4.32[12]: The total bondage number of the complete bipartite graph K m, n with
2 ≤ m ≤ n is b t(K m, n) = m.
Theorem 4.33 [12]: The total bondage number of a wheel of order p ≥ 5 is
b t( W p ) = 2.
Also Kulli and Patwari [12] established bounds of total bondage number
Theorem 4.32 [12]: If G is a connected graph of order p ≥ 5, then b t(G) ≤≤≤≤ p-5.
Theorem 4.33 [12]: If T is a tree with at least 2 cut vertices, then b t(T) ≤≤≤≤ 2.
The following results are established by Jia Huang, Jun-Ming Xu Department of mathematics,
University of Science and Technology of China, Hefei, Anhui 230026, China.
Consider G = (V, E) as a digraph with the vertex-set V and the edge-set E. For a
subset S ⊂V, let E+(S) = {(u, v) ∈ E : u ∈S, v ∉ S} and E
−(S) = {(u, v) ∈ E : u ∉ S, v ∈ S}; let
N+(S) = {v ∈ V: ∃ u ∈ S, (u, v) ∈ E
+(S)} and N
−(S) = {u ∈ V: ∃ v ∈ S, (u, v) ∈ E
−(S)}. If S = {x}
we replace S by x for convenience. For v ∈ V and (u, v), (v,w) ∈ E, u and w are called an in-
neighbor and an out-neighbor of v, respectively . The in-degree and out-degree of v are the number
98
of its in-neighbors and out-neighbors, and are denoted by d−(v) = d
−G (v) and d
+(v) = d
+ G
(v), respectively. The degree of v is d (v) = d G(v) = d+(v) + d
−(v). Denote the maximum and the
minimum degree of G by ∆(G) and δ(G), the maximum and the minimum in degree (resp. out-
degree) of G by ∆− (G) and δ−
(G) (resp. ∆+ (G) and δ+
(G)). A digraph G is d-regular if δ− (G) = δ+
(G) = ∆−(G) = ∆+
(G) = d.
Lemma 4.34: Let G be a loopless digraph with δ−(G) ≥ 2. If there exist an edge (u, v) and a vertex
w different from u and v in G such that X = (N−(v) ∩ N
−(w)) \ {u} ≠ ∅ then
(a) b t (G) ≤ d+(u) + d
−(v) + d
−(w) − |N
−(v) \ N
−(w)| − 2;
(b) b t (G) ≤ d+(u) + d
−(v) + d
−(w) – min x∈X {|N
+(u) ∩ N
+(x)|} − 2.
Corollary 4.35: Let G be a loopless digraph. If δ+ (G) ≥ 2 and δ−
(G) ≥ 2, then
b t (G) ≤ min{δ + (G) + 2 ∆−(G) , δ−
(G) + ∆+(G) +∆−
(G)} − 3.
In particular, if G is δ-regular and δ≥ 2, then b t (G) ≤ 3(δ − 1).
Lemma 4.36: For a digraph G, bt (G) ≥ s(G).
Denote by s(G) the minimum number of edges which support all minimum total dominating
sets in G.
Split Bondage Number: b s (G):
The removal of set of edges of G results a spanning subgraph H of G such that the split
domination number of H may be greater than or less than the split domination number of G. This
motivates to define new parameter as “The Split bondage number”. These concepts were
introduced by Kulli, Janakiram and Iyer [10].
4.4.3 Split Bondage Number: b s (G):
The Split bondage number b s(G) of a graph G is the minimum cardinality among
all sets of edges X ⊂ E such that γ s(G-X) > γ s(G).
Ex.c6
99
Clearly γ s(G-X) > γ s(G).
b s(G) = 2
Negative Split Bondage Number: b −−−− s (G):
The negative split bondage number b −−−− s (G) of a graph G is the minimum
cardinality among all sets of edges X ⊂ E such that γ s(G-X) < γ s(G).
Theorem 4.37 [10]: For any connected graph G, b s (G) ≤≤≤≤ p -1.
Theorem 4.38 [10]: If γ(G) = γ s(G), then b s(G) ≤ b (G).
Kulli and Janakiram [10] obtained an upper bound on b − s (G) in terms of the maximum
degree ∆(G).
Theorem 4.39 [10]: For any connected graph G, b s(G) ≤ ∆(G) – 1.
Theorem 4.39 [10]: Let G be a graph with diam (G) = 2 and D be γ s set of G. If there exists a
b −−−− s set X in ⟨V −D⟩ incident to some vertex v ∈ V-D, then γ s(G) ≤ b − s (G) + ∆(G) – 1
Theorem 4.40 [10]: If v is a vertex of minimum degree 2 and u, w are two adjacent vertices
adjacent to v, then b − s (G) = 1.
Definition: A graph is said to be well dominated if and only if every minimal dominating set is
of same cardinality [11].
Theorem 4.41 [10]: If G is a well dominated graph with γ s(G) = α0 (G), then b − s (G) = 1.
Theorem 4.42 [10]: If G is a well dominated graph, then b − s (G) ≤ p - γ s(G).
Edge Removal: Unchanging Domination
100
Consider a graph for which the domination number is unchanged when any single edge is
removed. These graphs were characterised by Walikar and Acharya [5].
Theorem 4.43 [5]: A graph G has γ(G) = γ(G-e) for any edge e ∈ E if and only if for each
e = uv ∈ E, there exists a γ set D such that one of the following condition is satisfied:
(a) u, v ∈ D
(b) u, v ∈ V – D
(c ) u ∈ D and v ∈ V-D implies |N(v) ∩ D | ≥ 2.
Nonbondage Number: This is used in a communication network. That is it is used to minimize
the direct communication links in the network. Kulli and Janakiram [9] introduced
Nonbondage number.
Nonbondage Number: The nonbondage number b n (G) of a graph G is the maximum
cardinality among all sets of edges X ⊆ E such that γ (G-X) = γ (G).
γ (G) = 2 γ (G-X) = 2
b n (G) = 5
Kulli and Janakiram established the following theorems and corollaries.
Theorem 4.44[9]: For any graph G, b n (G) = q – p + γ (G).
101
Here q = 11, p = 8 and γ (G) = 2
b n (G) = 11 - 8 +2 = 5
Corollary 4.45 [9]: For any graph G, b n (G) ≤ q – ∆ (G).
Here q = 11, ∆ (G) = 5 and b n (G) = 5
Clearly b n (G) ≤ q – ∆ (G).
Corollary 4.45 [9]: For any connected graph G, (diam (G) – 2)/3 ≤ b n (G).
Corollary 4.46 [9]: If G is a Hamiltonian graph, then p/3 ≤ b n (G).
Theorem 4.47[9]: For any graph G, b (G) ≤ b n (G) + 1.
Corollary 4.48 [9]: For any graph G, b (G) ≤ q - ∆(G) + 1.
102
Theorem 4.49[9]: Let G be a unicyclic graph. If γ (G) = p/2, then ∆(G) ≤ b n (G).
Theorem 4.50[9]: An edge e = uv is in every b n – set of G if and only if for every γ set D of G,
{u,v} ⊆ D or {u,v} ⊆ V – D.
Edge addition: Changing Domination:
The domination number is changed when any single edge is added....
i.e. γ(G + e) = γ(G) - 1 .
Ex.
γ(G) = 3 γ(G + e) = 2
γ(G + e) = γ(G) - 1 .
The addition of an edge can decrease the domination number by at most one. Sumner and
Blitch [13] characterised these graphs only in the cases for which γ (G) is 1 or 2.
Theorem 4.51[13]: A graph G with γ (G) =1 is in CEA if and only if G is complete.
Theorem 4.52[13]: A graph G with γ (G) =2 is in CEA if and only if G is complement of a
union of stars.
Sumner [14] characterized the disconnected graphs in CEA having γ (G) = 3.
103
Theorem 4.53 [14]: A disconnected graph G with γ (G) = 3 is in CEA if and only if G A ∪ B
where either A is trivial and B is in CEA and has γ (G) = 2 or A is a complete graph and B is a
complete graph minus a 1-factor.
Although, the graphs in CEA with γ (G) ≥ 3 have not been characterized, many interesting
properties of these graphs have been found. For example, Favaron, Sumner and Wojcicka [15]
showed the following result.
Theorem 4.54 [14]: The diameter of a graph G∈CEA with γ (G) = k is at most 2k-2.
They conjectured that the best possible bound is actually 3k/2 – 1 . This conjecture is
unsolved at the present time.
Sumner and Blitch [13] also conjectured that all graphs in CEA have equal domination and
independent domination numbers. This conjecture is true for graphs in CEA which have γ(G) ≤2
Ao, Cockayne, MacGillivray, and Mynhardt constructed a counter example for graphs G in
CEA with γ (G) ≥ 4. The conjecture is still unsettled for graph G∈CEA with γ (G) = 3, and many
people who have studied it believe it is true for this case.
Wojcicka [15] proved another conjecture by Sumner and Blitch [13] that every connected
graph in CEA with γ (G) = 3 and p ≥ 6 has a Hamiltonian path.
Relating edge addition to vertex removal, Sumner and Blitch [13] showed that V+
is empty for
graphs in CEA.
Theorem 4.55 [13]: If a graph G∈ CEA, then V = V− ∪ V
0.
Favaron et al. gave another property of the vertex set of graph in CEA.
Theorem 4.56 [15]: If a graph G∈ CEA, then the sub graph induced by V0 is complete graph.
Theorem 4.57 [15]: If a connected graphs G ∈ CEA is not complete then | V− | ≥ γ (G).
Cobondage Number: The cobondage number b c (G) of a graph G is the minimum cardinality
among the sets of edges X ⊆ P2(V) – E, Where P2(V) = { X ⊆ V: |X| = 2 } such that γ (G + X) <
γ (G).
104
Ex. G:
γ (G) = 2
Here E = {AB, BD, DC, AC} and P2(V) = { AB, BD, DC, AC, AD, BC}
X ⊆ { AD, BC}
γ (G + X) = 1
γ (G + X) < γ (G).
b c (G) = 1
This concept was defined by Kulli and Janakiram [16].
An obvious application of this theory is evident in our communication network example.
Determining b c (G) for the networks underlying graph reveals the minimum number of
communication links which must be added to the network in order to decrease the number of file
servers required to service the system.
Kok and Mynhardt [17] (the term “reinforcement number” was used instead of cobondage
number) found cobondage number for several families of graphs and determined bounds on
cobondage number.
Theorem 4.58 [17]: For any graph G, γ (G) ≤ p - ∆(G) - b c (G) + 1.
Theorem 4.59 [17]: If G is a graph with γ (G) ≥ 2 then b c (G) = µ(G).
Corollary 4.60 [17]: If G is a graph with γ (G) = 2 then b c (G) = p - ∆(G) -1.
105
Kulli and Janakiram [16] established an upper bound on bondage number of a complementary
graph.
Theorem 4.61 [16]: For any graph G, b c ( ) ≤ δ (G).
Theorem 4.62 [16]: For any graph G, b c (G) ≤ ∆ (G) + 1.
Theorem 4.63 [16]: For any graph G, b c (G) ≤ ∆ (G) + 1 if and only if every γ – set of G
satisfying the following conditions:
(i) D is independent,
(ii) every vertex in D is of maximum degree,
(iii) every vertex in V – D is adjacent to exactly one vertex in D.
Theorem 4.64 [16]: For any graph G, b c (G) ≤ p – 1 with equality if and only if G = 2.
Theorem 4.65 [16]: For any graph G of order p ≥ 3, b c (G) ≤ p – 2 with equality if and only if
G = 2 K2 or 3 or K2 ∪K1.
The following results gave some relationships between b c (G) and γ (G).
Theorem 4.66 [16]: For any graph G, γ (G) + b c (G) ≤ p+1 with equality if and only if G = p.
Theorem 4.67 [16]: Let D be a γ - set of G. If there exists a vertex v∈D which is adjacent to
every other vertex in D, then γ (G) + b c (G) ≤ p-1.
Theorem 4.68 [16]: For any graph G, γ (G) + b c (G) ≤ p-∆(G) +1.
Theorem 4.69 [16]: If T is a spanning tree of G such that γ (T) =γ (G), then b c (G) ≤ b c (T).
Theorem 4.70 [16]: If T is a tree and u is a cut vertex adjacent to an end vertex, then
b c (T) ≤ 1 +min {deg(u)}.
A lower bound for b c (T) involving the bondage number of T which was given by Kulli and
Janakiram [16].
Theorem 4.71 [16]: if T is a tree with diam (T) = 5 and has exactly cut vertices which are
adjacent to end vertices and also they have same degree, then b (T) + 1 ≤ 1 + b c (T).
106
Kulli and Janakiram [16] determined the cobondage number for K n1, n2, n3, … , nt,
Preposition 4.72 [16]: if G = K n1, n2, n3, … , nt, where n1≤ n2 ≤n3 ≤…≤nt , then b c (G) = n1 – 1.
Kulli and Janakiram [16] and Kok and Mynhardt [17] determined the cobondage number for
paths and cycles by using different techniques.
Preposition 4.73 [16, 17]: If p = 3k + i ≥ 4 where i = 1, 2, 3 then b c (C n) = b c (P n) = i.
Definition: For a graph G, Let u∈D⊆V. The Private neighborhood of u with respect to D is the
set pn [u, D] = N[u] – N[D-{u}]. Let p(D) = min{|pn[u, D] |: u ∈ D}. The private neighborhood
number p(G) of G is the min{p(D): D is a γ-set of G}.
Kok and Mynhardt [17] showed that the cobondage number is bounded by the private
neighborhood number.
Theorem 4.74 [16]: If G is a graph with γ (G) ≥ 2, then b c (G) ≤ p(G) with equality holds if
b c (G) = 1.
Edge addition: Unchanging Domination (UEA)
The domination number is unchanged when any single edge is added.
γ(G + e) = γ(G)
Ex.
γ(G) = 2 γ (G + e) = 2
107
γ(G + e) = γ(G).
Theorem 4.75 [3]: A graph G ∈ UEA if and only if V−
is empty.
Observation
Classes of changing and unchanging graphs:
The Venn diagram in the following figure illustrates the relationships among the
classes. Not all graphs are in one of the six classes.
� Graph G ∈ (UVR ∩ UEA ) − (UER ∪ CER)
� Graph G ∈ UER − (CVR ∪ CEA ∪ UEA)
108
� Graph G ∈ (UER ∩ UEA ) − UVR Also this graph has V = V0∪V
+
Conclusion: Relationships among Classes:
There are many interesting relationships among the six classes of changing and
unchanging graphs. For example, the characterization of the graphs in UEA relates them to the
graphs in CVR.
Observation 4.76:
(a) A graph G∈ UVR if and only if V = V0.
(b) If a graph G∈ UER, then V = V 0 U V
− U V
+-
(c) A graph G∈ UEA if and only if V = V 0 U V
+- (either V
0 or V
+ may be
empty).
(d) A graph G∈ CVR if and only if V = V−
(e) If a graph G∈ UVR, then G ∈ UEA.
(f) If a graph G ∈ CER ∩ UVR if and only if G is mK2 m ≥2.
109
(g) A graph G ∈ (CER ∩ UEA) – UVR if and only if G is a galaxy with no isolated
vertices and at least one star with two or more end vertices.
(h) A graph G ∈ CER −(UEA ∪ CEA) if and only if G is a galaxy with at least one
isolated vertex and at least two edges.
(i) A graph G ∈ CER ∩ CEA if and only if G has n ≥ 3 vertices and exactly one
edge.
(j) If a graph G ∈ CVR, then G ∈ UER.
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122
A (or AG) Adjacency matrix, 25 b(G) Bondage number, 88 Bn. Bouquet, 13 CLn Circular ladder graph, 14 γ cl (G) Clique domination number, 62 ω (G) Clique number, 23 Complement of a graph, 23
Km,n. Complete bipartite graph, 17 Kn Complete graph, 14 γ c (G) Connected dominating number, 60
γ c 1(G): Connected edge domination number, 66
γ ct (G) Connected total domination number, 61 γ cot (G) Cototal domination number, 76
γ cns (G) Cycle non split domination number, 74 Cn Cycle, 19 deg (v) Degree of vertex (valency), 7 diam(G) Diameter, 12
Dn Dipole, 13 d2–dom(G) = 2
Distance-k domination number, 77 d(G) Domatic number, 67 D Dominating set, 56 γ (G) Domination number, 56 b1 (G) Edge independence number, 35 k1 (G) Edge covering number, 35 γ
1(G) Edge domination number, 65
λ(G) Edge-connectivity, 16 d(u , v) Geodesic distance, 12 g (G) Girth, 19 γ g (G) Global domination number, 64
123
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
Wn Wheel, 18
124
125
Adjacency matrix, 19
Alternating path, 37
Augmenting path, 37
Biclique, 15
Bigraph, 17
Bipartite graph, 17
Block 16
Block graph 16
Bondage number, 88
Bouquet, 12
Bridge, 15
Circuit, 10
Circular ladder graph, 14
Clique dominating set, 62
Clique domination number, 62
Clique, 23
Complement, 23
Complete bipartite graph, 17
Complete graph, 14
Components, 11
Connected dominating number, 60
Connected dominating set, 60
Connected edge dominating set, 66
Connected edge domination number, 66
Connected graph, 12
Connected total dominating set, 61
Connected total domination number, 61
Cototal dominating set, 76
Cototal domination number, 76
Cut point, 14
126
Cycle non split dominating set, 74
Cycle non split domination number, 74
Cycle, 19
Degree of vertex, 7
Density, 12
Diameter, 12
Digraph, 5
Dipole, 13
Directed graph , 4
Distance-k dominating set, 77
Domatic number, 67
Dominating set, 56
Domination number, 56
Eccentricity, 12
Edge cover, 35
Edge covering number, 35
Edge deleted subgraph, 22
Edge dominating set, 65
Edge domination number, 65
Edge independence number, 35
Edge-connectivity, 16
Edge-induced subgraph, 22
Euler tour and hamilton cycles, 28
Factorization, 37
Forest, 27
General graph, 6
Geodesic distance, 12
Girth, 19
Global dominating set, 64
Global domination number, 64
Graph, 4
Incidence matrix, 20
Independence number, 33
Independent dominating set, 58
Independent domination number, 58
Independent set of edges, 34
Independent set, 33
Induced paired dominating set, 63
Induced paired domination number, 63
Induced subgraph, 21
127
Isomorphism of graphs, 21
Loop, 5
Matching, 35
Maximal independent set, 33
Maximum independent set, 33
Maximum matching, 36
Minimal dominating set, 57
Minimum dominating set 57
Multigraph, 6
Multiple edges, 5
Negative split bondage number, 98
Non split dominating set, 73
Non split domination number, 73
Null graph, 5
Operation on graph theory, 31
Order of the graph, 5
Paired dominating set, 62
Paired domination number, 62
Path graph 14
Path non split dominating set, 75
Path non split domination number, 75
Path, 9
Perfect matching , 36
Petersen graph 9
Planar graph, 18
Point cover, 35
Proper subgraph, 21
Pseudo graph, 6
Radius, 12
Regular graph, 8
Simple graph, 6
Size of the graph, 5
Spanning subgraph, 21
Spanning tree, 27
Split bondage number, 97
Split dominating set, 72
Split domination number, 72
Stable set, 33
Star, 18
Strong split dominating set, 72
128
Strong split domination number, 73
Subgraphs, 20
Distance-k domination number, 77
Nonbondage number, 99
Total bondage number, 93
Total dominating number, 59
Total dominating set, 59
Total edge dominating set, 65
Total edge domination number, 65
Total global dominating set, 64
Total global domination number, 64
Trail, 9
Tree, 27
Trivial graph, 5
Undirected graph, 4
Unicyclic graph, 20
Valency, 7
Vertex covering number, 35
Vertex deleted subgraph, 22
Vertex-connectivity, 15
Vertex-induced subgraph, 21
Walk, 9
Wheel, 18
