Table of Contents
ISRN Combinatorics
Volume 2013, Article ID 201654, 4 pages
http://dx.doi.org/10.1155/2013/201654
Research Article

Some New Perspectives on Global Domination in Graphs

1Saurashtra University, Rajkot, Gujarat 360005, India
2A. V. Parekh Technical Institute, Rajkot, Gujarat 360001, India

Received 10 May 2013; Accepted 20 July 2013

Academic Editors: P. Garcia-Vazquez and P. E. Jorgensen

Copyright © 2013 S. K. Vaidya and R. M. Pandit. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A dominating set is called a global dominating set if it is a dominating set of a graph and its complement . Here we explore the possibility to relate the domination number of graph and the global domination number of the larger graph obtained from by means of various graph operations. In this paper we consider the following problem: Does the global domination number remain invariant under any graph operations? We present an affirmative answer to this problem and establish several results.

1. Introduction

The domination in graphs is one of the concepts in graph theory which has attracted many researchers to work on it. Many variants of dominating sets are available in the existing literature. This paper is focused on global domination in graphs.

We begin with simple, finite, and undirected graph with . The set is called a dominating set if . A dominating set is called a minimal dominating set (MDS) if no proper subset of is a dominating set.

The minimum cardinality of a dominating set in is called the domination number of denoted by , and the corresponding dominating set is called a -set of .

The complement of is the graph with vertex set in which two vertices are adjacent in if and only if they are not neighbors in .

A dominating set of is called a global dominating set if it is also a dominating set of . The global domination number is the minimum cardinality of a global dominating set of . The concept of global domination in a graph was introduced by Sampathkumar [1]. This concept is remained in focus of many researchers. For example, the global domination number of boolean function graph is discussed by Janakiraman et al. [2]. The NP completeness of global domination problems is discussed by Carrington [3] and by Carrington and Brigham [4]. The global domination number for the larger graphs obtained from the given graph is discussed by Vaidya and Pandit [5] while Kulli and Janakiram [6] have introduced the concept of total global dominating sets. The discussion on global domination in graphs of small diameters is carried out by Gangadharappa and Desai [7].

The wheel is defined to be the join where . The vertex corresponding to is known as apex vertex, and the vertices corresponding to cycle are known as rim vertices.

Duplication of an edge of a graph produces a new graph by adding an edge such that and .

The shadow graph of a connected graph is constructed by taking two copies of , say and . Join each vertex in to the neighbors of the corresponding vertex in .

A vertex switching of a graph is the graph obtained by taking a vertex of , removing all the edges incident to and adding edges joining to every vertex not adjacent to in .

For the various graph theoretic notations and terminology we follow West [8] while the terms related to the concept of domination are used in the sense of Haynes et al. [9].

Here we consider the problem: Does the global domination number remain invariant under any graph operations? We present here an affirmative answer to this question for the graphs obtained by various graph operations on and . Moreover, we obtain the domination number and the global domination number for the shadow graph of and the graphs obtained by switching of a vertex in as well as in .

2. Main Results

Theorem 1. If is a graph obtained by duplication of an edge in by an edge then every -set of is a global dominating set of and . Furthermore, the global domination number remains invariant under the operation of duplication of an edge for .

Proof. Let be a graph obtained by duplication of an edge in . Without loss of generality, let the edge of be duplicated by an edge .
Let be a -set of . Then is a dominating set of . Now, any two distinct vertices and with in are enough to dominate because the vertices which are not in must belong to in . Since every -set of contains at least two such vertices and , it is a dominating set of . Hence, every -set of is a dominating set of as well as of . Consequently, every -set of is a global dominating set of .
Now, consider a -set of :
Since , must contain and for minimum cardinality. Also, to retain the minimum cardinality of , it must contain the vertices . That is, . Now, being a -set of is a global dominating set of with minimum cardinality . This implies that . Also, as reported in Sampathkumar [1], . Hence, . Thus, the global domination number remains invariant under the operation of duplication of an edge by an edge in .

Theorem 2. If is the graph obtained by duplicating each edge of by an edge then every -set of is a global dominating set of and .

Proof. Let be a -set of . Then is a dominating set of . Now, any two adjacent vertices and in are enough to dominate because the vertices which are not in in must belong to in . Since every -set of contains at least two such vertices and , every -set of is a dominating set of . Hence, every -set of is a dominating set of as well as of . This implies that every -set of is a global dominating set of .
Let be the vertices of . Consider a -set of , . being a -set of is a dominating set of with minimum cardinality. Moreover, since are the vertices of maximum degree in and from the nature of the graph , it is clear that is of minimum cardinality. Hence, is a -set of with minimum cardinality . Therefore, . Now, being a -set of is a global dominating set of with minimum cardinality which implies that as required.

The following Theorem 3 can be proved by the arguments analogous to the above Theorem 2.

Theorem 3. If is the graph obtained by duplicating each edge of by an edge then every -set of is a global dominating set of and .

Theorem 4. If is a graph obtained by duplicating an edge of by an edge then
That is, the global domination number remains invariant under the operation of duplication of an edge in .

Proof. Let , where is the apex vertex of .
It is easy to observe that
Case I (When a rim edge of is duplicated by an edge). Without loss of generality, let the rim edge of be duplicated by an edge .
For , since the vertex is adjacent to each vertex of , it must belong to any global dominating set of . Moreover, any two vertices are adjacent to third vertex other than in . Therefore, any global dominating set of must contain at least four vertices including which implies that .
For , the vertex dominates while the vertex and any two adjacent rim vertices of are enough to dominate . Therefore, any global dominating set of must contain at least three vertices of . This shows that .
Thus,
Case II (When a spoke edge of is duplicated by an edge). Without loss of generality, let the spoke edge be duplicated by an edge .
For , any two vertices in are adjacent to the third vertex and also any three vertices are adjacent to the fourth vertex in . Therefore, any global dominating set of must contain at least four vertices which implies that .
For , clearly any global dominating set of must contain either or to achieve its minimum cardinality. Moreover, any two adjacent rim vertices of and the vertex are enough to dominate and they also dominate . This implies that .
Thus,
Hence, we have proved that
That is, the global domination number remains invariant under the operation of duplication of an edge in .

Theorem 5. Every -set of is a global dominating set of , and

Proof. Consider two copies of . Let be the vertices of the first copy of and the vertices of the second copy of .
If is a -set of then is a dominating set of . Now, any two adjacent vertices and of are enough to dominate because the vertices which are not in in must belong to in . Since every -set of contains at least two such vertices and , every -set of is a dominating set of . Hence, every -set of is a dominating set of as well as of . This implies that every -set of is a global dominating set of .
Case I . For and , clearly and are -sets as well as global dominating sets with minimum cardinality respectively. Therefore, for .
Case II . (i) For i.e., , consider a -set where .
(ii) For i.e., , consider a -set where .
(iii) For i.e., or , consider a -set where .
Now, being a -set of is a global dominating set of with minimum cardinality implying that .
Thus, we have proved that

Theorem 6. If is a graph obtained by switching of a vertex in cycle then .

Proof. Let be the successive vertices of and denotes the graph obtained by switching of a vertex of . Without loss of generality, let the switched vertex be .
Consider a set . Then is a dominating set of as all the vertices except the pendant vertices, namely, and , are in while and are already in . Moreover, the set is a minimal dominating set of because for any , the set does not dominate the vertex of . Furthermore, the vertex dominates vertices of and the remaining two vertices are pendant vertices. Therefore, at least three vertices are required to dominate , and hence . Thus, being a minimal dominating set with minimum cardinality is a -set of which implies that .
Now, we claim that the pendant vertices in are enough to dominate remaining vertices of . Since the vertex which is not in in must belong to in , any containing and will be a dominating set of . Thus, is a dominating set of as well as of . This implies that is a global dominating set of .
Since is a -set of as above, it is of minimum cardinality. Therefore, . Thus, as required.

Theorem 7. If is a graph obtained by switching of a rim vertex in a wheel then

Proof. Let be the successive rim vertices of and denotes the graph obtained by switching of the vertex of . Without loss of generality, let the switched vertex be . Let be the apex vertex of .
Case I . It is easy to observe that is a global dominating set of with minimum cardinality. Therefore, for .
Case II . Consider a set . Then, is a dominating set of as all the vertices except the vertex are in in while is dominated by . Now, the vertex dominates all the vertices of except the vertices , and . But the vertices and are dominated by the vertices and in respectively while the vertices and are already in . Therefore, is a dominating set of . Hence, is a dominating set of as well as of . That is, is a global dominating set of . Moreover, two vertices, namely, and either of the vertices from and are enough to dominate . Since any two vertices in are adjacent to the third vertex in , at least three vertices are essential to dominate . This implies that . Thus, we have proved that

Theorem 8. If is the graph obtained by switching of the apex vertex in wheel then .

Proof. Let be the successive rim vertices of and let denote the graph obtained by switching of the vertex of . Let the switched vertex be the apex vertex of .
Because is adjacent to every other vertex in , is not adjacent to any other vertex in . Therefore, any dominating set for must contain . This implies that any global dominating set of must contain .
Case I . It is easy to observe that and are -sets as well as global dominating sets of of and respectively with minimum cardinality.
Therefore,
Thus, for .
Case II . The vertex will be the isolated vertex in . Hence, . Therefore, . But as reported in Sampathkumar [1], . This implies that .
Because every dominating set of must contain and is enough to dominate , .
Hence, for .
Thus, we have proved that as required.

3. Concluding Remarks

The concept of global domination is remarkable as it relates the dominating sets of a graph and its complement. We have explored this concept in the context of some graph operations and also have investigated the domination number and the global domination number for the larger graph obtained by some graph operations on a given graph. The invariance parameter for global domination is also explored.

Acknowledgment

The authors are highly thankful to the anonymous referees for their kind comments and fruitful suggestions on the first draft of this paper.

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