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 ,