Chapter 1

Introduction

1

1.1 Introduction

The present thesis entitled A Study on Graph Transformations

and Their Relations with Graph Theoretical Parameters in The-

ory of Graphs is in the field of Graph Theory, which is one of the ever

growing branch of Mathematics. This growth is due to its wide applica-

tions to the problems originating from various fields like engineering, social

and biological sciences, economics, chemistry, theoretical physics, com-

munication networks, linguistics, discrete optimization problems, combi-

natorial problems and classical algebraic problems. This thesis focuses

mainly on the study of graph transformations and their relations like

covering, edge connectivity, traversability, nonplanarity, outerplanarity,

crossing numbers, edge decomposition and different domination parame-

ters in graph valued functions and also relations with graph theoretical

parameters in theory of graphs.

1.2 A brief history of Graph Theory

Unlike other areas in Mathematics, the theory of graphs has a def-

inite starting place, a paper Solutio Problematis ad Geometriam Situs

2

Pertinentis [30] published in 1736 by the Swiss Mathematician Leonhard

Euler(1707-1783). The main idea behind this work grew out of a popular

problem known as Konigsberg Bridge Problem: The city of Konigsberg

was located on the Pregel River in Prussia. The city occupied two islands

plus areas on both the banks. The citizens wondered whether they could

leave home, cross every bridge exactly once and return home. The famous

problem was converted to graph theoretical problem and was settled by

Euler through the above paper.

More than a century after Euler’s paper, Cayley was led by the study

of particular analytical forms arising from differential calculus to study

particular class of graphs called the trees. This study has many implica-

tions in theoretical chemistry. The involved techniques mainly concerned

the enumeration of graphs having particular properties. Cayley linked his

results on trees with the contemporary studies of chemical composition.

The fusion of the ideas coming from Mathematics with those coming from

chemistry is at the origin of a part of the standard terminology of graph

theory.

Firstly, since a graph is very convenient and natural way of representing

the relationship between objects. We represent objects by vertices and the

3

relationship between them by edges, if relationship exists between any two

objects(vertices) then we say they adjacent to each other. The term graph

was introduced by Sylvester in a paper [58] published in 1878 in Nature,

where he draws an analogy between quantic invariants and co-variants of

algebra and molecular diagrams.

The autonomous development of topology from 1860 to 1930, fertilized

graph theory back through the work of Jordan, Kuratowski and Whitney.

Another important factor of common development of graph theory and

topology came from the techniques of modern algebra. A later textbook

by Frank Harary [32], published in 1969, was enormously popular and en-

abled mathematicians, chemists, electrical engineers and social scientists

to talk each other.

1.3 A brief introduction of graph valued functions

and graph transformations

In management and social networks, the incident and non-incident re-

lation of vertices and edges or adjacency and non-adjacency of vertices or

edges are used to define different problem in networks. Some times, it is

not possible to retrieve the original graph from the transformation graph

4

in polynomial time. Hence these graph transformations may be used in

graph coding or coding of some grouped signals. Also it is convenient to

study the structure of original graphs through the graph transformations.

This is the motivation behind to study the structure of exploration of

various graph transformations and study of their structural properties.

Defining a new graph from a given graph, by using incident relationship

between vertices and edges and adjacency relation between two vertices or

two edges is known as graph transformation. It transforms a new structure

from the original graph either by adding an extra information [vertices or

edges] or extracting a new information from the substructure by involving

the above cases of the original graph.

We begin with some graph valued functions of a graph which are the

most interesting operations by which one graph is obtained from the other.

Firstly, the complement of a graph G is denoted by G , is a graph with

vertex set V (G) in which two vertices are adjacent if and only if they are

not adjacent in G . If x = uv is an edge of G , and w is not a vertex of

G , then x is subdivided when it is replaced by the edges uw and wv . If

every edge of G is subdivided, the resulting graph is known as the subdi-

vision graph S(G) or simply S(G) is obtained by inserting a new vertex

5

in each edge of G . Now the graph valued function which has received

the most attention, is the concept of line graph, first studied by Whitney

[60] and surprisingly discovered independently by many graph theorists.

A line graph denoted by L(G) of a graph G , is the graph with vertex

set as the edge set of G and two vertices of L(G) are adjacent whenever

the corresponding edges in G have a vertex in common. The jump graph

J(G) of a graph G is a graph whose vertex set is the edge set of G and

two vertices of J(G) are adjacent if and only if the corresponding edges

in G are not adjacent in G . The jump graph J(G) of a graph G is

the complement of the line graph L(G) of a graph G . It was studied

by Chartrand [25]. One of the important variations of the line graph is

the middle graph, which has been introduced by Hamada and Yoshimura

[33]. The middle graph M(G) of a graph G , is a graph whose vertex set

is the union of vertex set and edge set of G , where two vertices of M(G)

are adjacent if and only if either they are adjacent edges of G or one is a

vertex and other is an edge of G incident with it. The concept of middle

graph M(G) of graph G was independently studied by Sampathkumar

and Chikkodimath, they called it as semitotal-line graph T1(G) of a graph

G and also in [55] they introduced the concept of semitotal-point graph

6

T2(G) of a graph G , which is a graph with vertex set V (G) ∪ E(G) in

which two vertices are adjacent if and only if they are adjacent vertices of

G or one is a vertex of G and other is an edge of G incident with it. The

block graph B(G) of a graph G was introduced by Harary [32], is a graph

with vertex set as a set of all blocks of G and two vertices of B(G) are

adjacent if and only if the corresponding blocks have the common vertex

in G . In[54] Ponaraj and Somasundaram have initiated a study on de-

gree splitting graph Ds(G) of graph G , which is defined as follows : Let

G = (V, E) be a graph with V = S1∪S2∪, ...,∪St∪T where each Si is a

set of vertices having at least two vertices and having the same degree and

T = V −∪Si . The degree splitting graph Ds(G) of a graph G is obtained

from G by adding vertices W1,W2, ....,Wt and joining Wi to each vertex

of Si (l ≤ i ≤ t) . Another variation in the line graph and middle graph

is a total graph. The total graph T (G) of a graph G is the graph whose

vertex set is V (G) ∪ E(G) and in which two vertices are adjacent if and

only if they are adjacent or incident in G . This concept was introduced

by Behzad [18]. The concept of total graph was generalized by Wu and

Meng [61], which is our aim to study the new graphical transformations

and is defined as follows: Here in a graph G = (V, E) , the vertices and

7

the edges of G are called the elements of G .

Let �, � be two elements of V (G)∪E(G) . We say that the associativ-

ity of � and � is + if they are adjacent or incident in G , otherwise − .

Let xyz be a 3-permutation of the set {+,−} . We say that � and �

correspond to the first term x (respectively to the second term y or the

third term z ) of xyz if both � and � are in V (G) (respectively both

� and � are in E(G) , or one of � and � is in V (G) and other is in

E(G) ). The transformation graph Gxyz of G is defined on the vertex set

V (G) ∪ E(G) . Two vertices � and � of Gxyz are joined by an edge if

and only if their associativity in G is consistent with the corresponding

terms of xyz .

Since there are eight distinct 3-permutations of +,− , we obtain eight

graphical transformations of G . It is interesting to see that G+++ is ex-

actly the total graph T (G) of G and G−−− is the complement of T (G) .

Also for a given graph G ,G++− and G−−+ , G−+− and G+−+ , G−++

and G+−− are the other three pair of complementary graphs.

8

1.4 Brief history of domination theory of graphs

The fastest growing area within graph theory is the study of domination

and related problems such as independence, covering and matching. The

historical root of domination has its origin in the game of chess, where the

aim is to cover(or dominate) various squares of a chessboard by certain

chess pieces, subject to variety of conditions.

In 1862 De Jaenisch considered dominating the squares with queens and

posed the following chess board problem: Determine the minimum num-

ber of queens that can be placed on a chessboard such that every square

is either occupied by a queen or can be occupied one of the queen in a

single move.

The minimum number of such queens is five. Using graph theory to

model this problem the Queen’s graph is formed by representing each of

the 64 squares of the chessboard as a vertex set of a graph G . Two ver-

tices(squares) are adjacent in G if each square can be reached by a queen

on the other square in a single move. Obviously, to solve the Queen’s

problem we are looking for the minimum number of queens that domi-

nate all the squares of a chessboard that is domination number. For a

comprehensive survey of chess board problem see [28], Cockayne et.al [27]

9

[29], Hedetniemi et.al [36], Spencer and Coackayne [57] and Wagner and

Geist [59].

The topic of domination began with Claude Berge [19] in 1958 and

Oystein Ore [51] in 1962, with Ore actually using the term ”domination”

for undirected graphs and also he introduced the concept of minimal and

minimum dominating sets of vertices in a graph. In 1977, Ernie Cock-

ayne of the University of Victoria and Stephen Hedetniemi of Clemson

University, published the first paper entitled [27], Towards a theory of

domination in graphs, after that domination became area of study by

many. In 1990, Hedetniemi and Laskar [37], published Bibliography on

domination in graphs and some basic definitions of domination parame-

ters, containing at that time about 400 entries. At the end of 1997, this

bibliography has grown to over 1200 entries. In 1998, a first large volume

text books Fundamentals of domination in graphs [34] and Domination

in graphs:Advanced topics [35] devoted to this area and the subject was

written and edited by Teresa Haynes of East Tennessee state University,

Heditniemi and Peter Slater of the University of Alabama in Huntsuille.

At the end of 2006, domination theory and its related parameters has

grown to more than 10000 entries in the different journals proceedings. A

10

recent book on domination by Kulli can be found in [48].

Hence in last three decades domination has emerged as one of the signif-

icant area of research not only in graph theory but also in combinatorial

optimization and analysis of algorithms.

1.5 Outline of the present investigation

The contents of this thesis is organized into eight chapters and are con-

veniently categorized into three parts. The first part consist of three

chapters from the beginning in which first Chapter deals brief history,

introduction, terminologies and definitions, which is introductory in na-

ture. In second Chapter we study basic properties, covering invariants,

connectivity and characterization of graphs whose transformation graph

G+−+ are eulerian. In Chapter three, we found the characterization

of graphs whose transformation graph G−++ are eulerian, outerplanar,

maximal outerplanar or minimally nonouterplanar and also we establish

a necessary and sufficient conditions for the transformation graphs G−++

to have a crossing number one or two.

The second part of the thesis contains two chapters, Chapter four

and five which deals with the edge decomposition of the transformation

11

graphs G++− and G+−− into some standard class of graphs.

Chapters 6,7 and 8 constitutes the third part of the thesis, which are

completely devoted to study the domination of transformation graph G+−+

and graph valued functions: semitotal-point graph and degree splitting

graph of a graph.

Chapter six deals with some results on domination number of transfor-

mation graph G+−+ of G and the upper bounds for G+−+ . In Chapter

seven, a domination parameter namely, co-total domination of semitotal-

point graph T2(G) of G is defined and many bounds of ct(T2(G)) in

terms of elements of G but not the elements of T2(G) are derived. Chap-

ter eight gives the variation in domination from the graph G to the

degree splitting graph Ds(G) and also bounds on domination number of

Ds(G) in terms of elements of G but not in terms of elements of Ds(G)

are obtained.

1.6 Basic Terminology and Definitions

This preliminary section is introduced in order to make the thesis self-

contained.

A graph G consists of a pair (V (G), E(G)) , where V (G) is the nonempty

12

finite set whose elements are called vertices (points) and E(G) is a set of

unordered pairs of distinct elements of V (G) . The elements of E(G) are

called edges (lines) of the graph G . Graphs discussed in this thesis are

undirected and simple. For graph theoretic terminology, we refer Harary

[32].

A graph with p vertices and q edges is called a (p, q) graph. The

(1, 0) graph is trivial. A graph with more than one vertex is a nontrivial

graph. The order of a graph G is the number of vertices in G and it is

denoted by ∣V (G)∣ . The size of a graph G is the number of edges in G

and is denoted by ∣E(G)∣ . When there is no possibility of confusion, we

write V (G) = V and E(G) = E . A graph H is said to be a subgraph

of a graph G if V (H) is a subset of V (G) and E(H) is a subset of

E(G) . If H is a subgraph of a graph G and V (H) = V (G) , then we

say that H is a spanning subgraph of G . For any subset S of V (G) , the

induced subgraph ⟨S⟩ is the maximal subgraph of G with vertex set S .

Two graphs G and H are isomorphic written as G ∼= H , if there exists a

one-to-one correspondence between their vertex sets which preserves adja-

cency and nonadjacency. The degree of a vertex v is the number of edges

of G incident with v and is denoted by degG(v) or simply deg(v) . A

13

vertex of degree zero in G is called an isolated vertex. A vertex of degree

one in G is called a pendantvertex. A pendantedge is an edge incident to

a pendantvertex. The edge degree of an edge x=uv of a graph G is the

sum of the degrees of u and v. The minimum degree among all the ver-

tices(edges) of G is denoted by �(G) ( �′(G) ) and Δ(G) (Δ′(G)) denotes

the maximum vertex(edge) degree of G . If �(G) = Δ(G) , then the graph

G is said to be regular. Let a, b be any positive integer, 1 ≤ a ≤ b . A

graph G is said to be (a, b) -biregular or simply biregular if its vertices

have degree either a or b .

A walk is defined as a finite alternating sequence of vertices and edges,

beginning and ending with vertices, in which each edge is incident with

the two vertices immediately preceding and following it. A trail is a walk

in which all the edges are distinct and it is a path if all the vertices are

distinct. A path with p vertices is denoted by Pp . A closed path is called

a cycle. A cycle with p vertices is denoted by Cp . The length of a cycle

or a path is the number of occurrences of edges in it. This term is unde-

fined if G has no cycles. The girth g(G) of a graph G , is the length of

shortest cycle (if any) in G . A graph is said to be connected if every pair

of its vertices are joined by a path. A graph which is not connected is said

14

to be disconnected. A graph whose edge set is empty is called a totally

disconnected graph. A maximal connected subgraph of a G is called a

connected component or simply a component of G . Thus a disconnected

graph has at least two components. A cutvertex of a graph G is a vertex

of G whose removal increases the number of components and a bridge is

such an edge. If v is a cutvertex (or x is a bridge) of a connected graph

G , then G− v (or G− x ) is disconnected. A nonseparable graph is con-

nected, nontrivial and has no cutvertices. A block of a graph is a maximal

nonseparable subgraph. If G is nonseparable, then G itself is called a

block. The distance d(u, v) between two vertices u and v in G is the

length of a shortest path joining them if any; otherwise d(u, v) = ∞ . A

shortest u−v path is called a geodesic. The diameter of a connected graph

G is the length of any longest geodesic and is denoted as diam(G) . The

eccentricity eccG(v) of a vertex v ∈ V (G) , is the largest distance between

v and all other vertices of G , i.e, eccG(v) = max{d(u, v)/∀u ∈ V (G)} .

A set S ⊆ V of vertices which covers all the edges of a graph G is called

a vertex cover of G . The smallest number of vertices in any vertex cover

for G is known as vertex covering number of a graph G and is denoted

by �0(G) . Similarly �1(G) is the smallest number of edges in any line

15

cover of G and is called edge covering number of graph G . A set of

vertices(edges) in G an independent set of vertices(edges) if no two of

them are adjacent. The largest number of vertices(edges) in such a set

is called the vertex(edge) independent number �0(G) (�1(G) ) of a graph

G . The vertex connectivity �(G) (edge connectivity �(G) ) of a graph

G is the minimum number of vertices(edges) whose removal results in a

disconnected or trivial graph.

A complete (p, q) graph is a p − 1 regular graph having p(p−1)2

edges

and is denoted by Kp . For p ≥ 4 , the wheel Wp is defined to be the

graph K1 +Cp−1 . A clique of a graph G is maximal complete subgraph.

A graph with cycles is called cyclic graph, otherwise acyclic. A tree is

a connected acyclic graph. Any graph without cycles is called forest. A

graph is said to be bipartite graph or bigraph if its vertex set V (G) can

be partitioned into two subsets V1 and V2 such that every edge of G

joins a vertex V1 with a vertex of V2 . If G contains every edge joining

V1 and V2 , then G is complete bipartite graph. If V1 and V2 have m

and n vertices, we write G = Km,n . A star is a complete bipartite graph

K1,n . A bistar is a graph obtained by joining the centers of two stars

with an edge similarly tristar is graph obtained by joining the centers of

16

three stars by two edges. The neighborhood of a vertex u in V is the set

N(u) consisting of all vertices v which are adjacent with u . The closed

neighborhood is N [u] = N(u) ∪ {u} .

To express the structure of a given graph in terms of smaller and simpler

graphs and to have a notational abbreviations for a graphs we use follow-

ing operations on graphs. The union of two graphs G1 and G2 denoted

by G1 ∪ G2 is a graph with vertex set V (G) = V (G1) ∪ V (G2) and the

edge set E(G) = E(G1) ∪ E(G2) . The join of two graphs is denoted by

G1 + G2 and is the union of G1 and G2 as well as all edges uv with

u ∈ V (G1) and v ∈ V (G2) . The cartesian product of the graphs G1 and

G2 denoted by G1 × G2 , is the graph with vertex set V (G1) × V (G2) ,

two vertices (u1, u2) and (v1, v2) being adjacent in G1 × G2 if and only

if either u1 = v1 and u2v2 ∈ E(G2) or u2 = v2 and u1v1 ∈ E(G1) . The

G2 corona of G1 is the graph G1 ∘G2 formed from one copy of G1 and

∣V (G1)∣ copies of G2 where the itℎ vertex of G1 is adjacent to every

vertex in the itℎ copy of G2 .

A set D ⊆ V (G) is said to be dominating set of G , if every vertex in

V (G) − D is adjacent to some vertex in D . D is said to be minimal

dominating set if D − {u} is not a dominating set for any u ∈ D . The

17

domination number (G) of G is the minimum cardinality of a minimal

dominating set. Different types of dominating sets have been studied by

imposing conditions on the dominating set.

Note: The other terminologies and different domination parameters are

defined as and when the need arises.

1.7 Applications

Graphs can be used to model many types of relations and process dynam-

ics in physical, biological and social systems. Many problems of practical

interest can be represented by graphs.

✠ Radar network: Berge in his book Graphs and Hyper graphs (1973),

mentions the problem of keeping all points in a network under the surveil-

lance of a set of radar stations. A number of strategic locations 1,2,3,...

called cells are kept under the surveillance of radar. suppose Radar in

cell 2 can survey the location 1,3 or 5. Similarly, 3 can be surveyed by

radar location 2 or 3 and so on. What is the minimum number of radar

stations needed to survey all locations? It is the domination number of

the network.

18

✠ Road networks: Suppose G is a graph representing a network of

roads connecting a number of locations. In such a network, a degree of

vertex v is the number of roads meeting at v . If deg(u) > deg(v) , then

naturally the traffic at u is heavier than that of v . If we consider the

traffic between u and v , preference should be given to the vehicles going

from u to v . Thus, in some sense, u strongly dominates v and v weakly

dominates u . These concepts are studied in Strong and Weak domina-

tions.

✠ Communication network: In designing a communication network,

it is important to give good connectivity among all sites at reasonable

cost. The reliability of communication networks is an important issue in

their design and operation. Telecommunication circuits become noisy and

unusable and the communication computers may also fail. Graph theory

is therefore useful in the study of the connectivity of communication net-

works, where a dominating set represents a set of cities which acting as a

transmitting stations, can transmit messages to every city in the network.

✠ Computer science: In computer science, graphs are used to repre-

sent networks of communications, data organization, computational de-

vices, the flow of computation etc. For example: The link structure of a

19

website could be represented by a directed graph. The vertices are the

web pages available at the website and directed edges from page A to B

exists if and only if A contains a link to B. The development of algorithms

to handle graphs is one of the major areas of interest in computer science.

✠ Linguistics: Since natural language often lends itself well to dis-

crete structure, graph theoretic methods in various forms, have proven

particularly useful in linguistics. Traditionally, syntax and computational

semantics follow tree based structures, whose expressive power lies in the

principle of compositionality, modelled in a hierarchical graphs. Seman-

tic networks are therefore important in computational linguistics. The

other methods in phonology (e.g. Optimality theory, which uses lattice

graphs) and morphology (e.g. finite state morphology, using finite state

transducers) are common in the analysis of language as a graph. Indeed

the usefullness of this area of mathematics to linguistics has borne orga-

nizations such as TextGraphs, as well as various ’Net’ projects, such as

WordNet, VerbNet and others.

Domination is the one of the major areas of research in Graph theory.

The different domination problems serves as an answer to different real

life applications like different versions of watchman problem, radio station

20

communication, radar station location, coding theory, computer network-

ing etc.

✠ Chemistry: In chemistry, a graph makes a natural model for a

molecule, where vertices represent atoms and edges represents bonds.

Graph enumeration can be used to distinguish the different isomers of

a molecule. Graph theoretic approach is especially used in computer pro-

cessing of molecular structures, ranging from chemical editors to database

searching. Also graph theoretic methods are useful in finding the energy

of atoms and studies on it.

21