International Journal of Combinatorics

Volume 2013 (2013), Article ID 795401, 4 pages

http://dx.doi.org/10.1155/2013/795401

## Some New Results on Distance -Domination in Graphs

^{1}Department of Mathematics, Saurashtra University, Rajkot, Gujarat 360005, India^{2}B. H. Gardi College of Engineering & Technology, P.O. Box 215, Rajkot, Gujarat 360001, India

Received 17 August 2013; Revised 9 December 2013; Accepted 11 December 2013

Academic Editor: Toufik Mansour

Copyright © 2013 Samir K. Vaidya and Nirang J. Kothari. 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

We determine the distance -domination number for the total graph, shadow graph, and middle graph of path .

#### 1. Introduction

We begin with finite, connected, and undirected graphs, without loops or multiple edges. A dominating set of a graph is a set of vertices of such that every vertex of is adjacent to some vertex of . The domination number is the minimum cardinality of a dominating set of . Further, the open neighbourhood of is the set . The closed neighbourhood of is the set . The distance between two vertices and is the length of shortest path between and in , if exists otherwise, . The open -neighbourhood set of vertex is the set of all vertices of which are different from and at distance at most from in , that is, . The closed -neighbourhood set of is defined as . Obviously .

The *total graph * of a graph is the graph whose vertex set is and two vertices are adjacent whenever they are either adjacent or incident in .

The *Shadow graph * of a connected graph is obtained by taking two copies of , say and . Join each vertex in to the neighbours of corresponding vertex in .

The middle graph of a graph is the graph whose vertex set is and in which two vertices are adjacent whenever either they are adjacent edges of or one is a vertex of and the other is an edge incident with it.

For standard terminology and notations we rely upon Balakrishnan and Ranganathan [1] and Haynes et al. [2].

The concept of distance dominating set was initiated by Slater [3] while the term distance -dominating set was coined by Henning et al. [4]. For an integer , a is a -dominating set of if every vertex in is within distance from some vertex . That is, . The minimum cardinality among all -dominating sets of is called the -domination number of and it is denoted by . It is obvious that . A -dominating set of cardinality is called a -set. The distance domination in the context of spanning tree is discussed by Griggs and Hutchinson [5] while bounds on the distance two-domination number and the classes of graphs attaining these bounds are reported in the work of Sridharan et al. [6]. In [7] Topp and Volkmann have discussed distance -domination as -covering and characterized connected graphs of order with distance -domination (-covering). Application of distance domination in Ad Hoc wireless networking is briefly discussed by Wu and Li [8]. More details and bibliographic references on distance -domination can be found in a survey paper by Henning [9].

#### 2. Some Definitions and Main Results

Proposition 1 (see [9]). *For , let be a -dominating set of a graph . Then is a minimal -dominating set of if and only if each has at least one of the following properties.*(1)*There exists a vertex ** such that **.*(2)*The vertex ** is at distance at least ** from every other vertex ** of ** in **.*

Theorem 2. *For , .*

*Proof. *Let and be the vertices and the edges of , respectively. Then will be the vertices of . Then is distance -dominating set of as . The set being a singleton set it is obviously a minimal distance -dominating set of .

Theorem 3. *For ,
*

*Proof. *Let and be the vertices and the edges of respectively. Then , will be the vertices of . Now every vertex from dominates vertices of ’s and vertices of ’s at a distance while every vertex from dominates vertices of ’s and vertices of ’s at a distance . Therefore at least one vertex from must belongs to any distance dominating set of .

Hence,

Now depending upon the number of vertices of , consider the following subsets.

For ,
for ,
for ,
We claim that each is a distance dominating set because
where , , and , where, .

Therefore
This implies that for . Now from the nature of , one can observe that every vertex of is at a distance at least apart from every other vertex of in .

Thus by Proposition 1, above defined is a minimal distance -dominating set of . Hence, from (2), for ,

Theorem 4. *For , .*

*Proof. *Consider two copies of path . Let be the vertices of first copy of path and be the vertices of second copy of path . Then is distance -dominating set as . The set being a singleton, set it is obviously a minimal distance -dominating set of .

Theorem 5. *For ,
*

*Proof. *Consider two copies of path . Let be the vertices of first copy of path and be the vertices of second copy of path . Now every vertex from dominates vertices of ’s and vertices of ’s at a distance while every vertex from dominates vertices of ’s and vertices of ’s at a distance . Therefore at least one vertex from must belongs to any distance -dominating set of .

Hence
Now depending upon the number of vertices of , consider the following subsets.

For ,
for ,
for ,
We claim that each is a distance -dominating set because
where , and , where .

Therefore
which implies that for any . Now from the nature of , one can observe that every vertex of is at a distance at least apart from every other vertex of in .

Thus by Proposition 1, above defined is a minimal distance -dominating set of . Hence from (10), for ,

Theorem 6. *For , .*

*Proof. *Let and be the vertices and the edges of , respectively. Then will be the vertices of . Then is distance -dominating set of as . The set being a singleton set, it is obviously a minimal distance -dominating set of .

Theorem 7. *For ,
*

*Proof. *Let and be the vertices and the edges of respectively. Then will be the vertices of . Now every vertex from dominates vertices of 's and vertices of 's at a distance while every vertex from dominates vertices of 's and vertices of 's at a distance . Therefore at least one vertex from must belong to any distance -dominating set of .

Hence,
Now depending upon the number of vertices of , consider the following subsets.

For ,
for ,
for ,
We claim that each is a distance -dominating set because
where , , and , where .

Therefore
This implies that for . Now from the nature of , one can observe that every vertex of is at a distance at least apart from every other vertex of in .

Thus by Proposition 1, above defined is a minimal distance -dominating set of . Hence from (18), for ,

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