Tikrit Journal of Pure Science

A dominating set S of a graph 𝐺 = (𝑉, 𝐸) , is a subset of the vertex set V (G) such that any vertex not in S is adjacent to at least one vertex in S .The domination number of a graph G denoted by 𝛾 (𝐺) is the minimum size of the dominating sets of G. In this paper we introduced the domination numbers of certain prism graphs.


Introduction
Throughout this paper we consider simple graphs, finite, undirected and contain no loops or multiple edges. Our terminology and notations will be standard except as indicated. For undefined terms see [1], [2] and [3]. For a graph G= (V, E) , V denotes its vertex set while E its edge set. If ⊆ then < > denotes the induced sub graph of G by the vertices of D. The cardinality of a set S denoted by | | is the number of elements of the S. A set ⊆ is said to be dominating set of G if every vertex in − is adjacent to some vertex in D. The cardinality of a minimum dominating set D is called the domination number of G and is denoted by ( ) [4]. In other words we defined the domination number ( ) of a graph G as the order of smallest dominating set of G. A dominating set S with | | = ( ) is called a minimum dominating set [5]. Dominating set appear to have their origins in the game of chess where the goal is to cover or dominate various squares of a chessboard by certain chess pieces. The problem of determining domination numbers of graphs first emerged in (1862) of De Jaenisch who wanted to find the minimal number of queens on a chessboard, such that every square is either occupied by a queen or can be reached by a queen with single move [6]. To date many papers have been written on domination in graphs like [6,7,8]. In additional domination has many applications like the problem of monitoring an electric power system [9]. The book by Haynes, Hedetniemi, and Slater [2] illustrates many interesting examples including dominating queens, set of representatives, school but routing, computer communication networks, radio stations, land surveying …etc. Among them, the classical problems of covering chessboards by the minimum number of chess pieces were important in stimulating the study of domination [10]. As usual we use⌊ ⌋ for the smallest integer not greater than x. Definition 1.1: A Prism graph , is a simple graph given by the Cartesian product graph , = × where is a cycle with m vertices and is a path with n vertices. It can therefore be viewed and formed by connecting n concentric cycle graphs along spokes. Also sometime , calls a circular ladder graph [11]. The following theorems have been used in this paper: Theorem 1.1 (see [12]): If G has no isolated vertices, then ( ) ≤ | ( )| 2 . Theorem 1.2 (see [8]): A dominating set D of a graph G is minimal if and only if for each vertex ∈ one of the following conditions satisfied (i) There exist a vertex ∈ − such that  , we give the dominating set for these two cases of ,2 by: = 1 ∪ 2 where 1 and 2 are the same sets in case (1).

On domination numbers of prism graph
The proof is similar to that in the case (1) for ∈ 1 , ∈ 2 and ∈ 3 so remained two    3 ( 4) give the set of dominating set of ,3 by the set of the vertices = 1 ∪ 2 ∪ 3 where 1 , 2 and 3 are the same sets in case (1).

On domination numbers of prism graph , :
In this section we first determine the domination number of 3,4 , 5,4 and 9,4 as aspecial cases then we determine the domination numbers of ,4 in general.

Lemma(1): ( 3,4 ) = 4
Proof: Let the vertices of this graph labeled as shown in Fig.4.    (1) and lemma (2) we have always v dominates itself or there is at least one vertex that adjacent to v not dominating with any vertex in ′ . So no proper subset ′ ⊆ dominating 9,4 . Hence   .Similarly for any chooses ∈ either in 1 or in 2 . For ∈ 1 , is dominates itself and all the vertices that adjacent to except when = (8 + ) we have three vertices of the form {8, (8 + 2 ), (9 + )} that adjacent to not dominating with any vertex in ′ . Also, for ∈ 2 , is dominates itself and all the vertices that adjacent to except when

On domination numbers of prism graph
, : As previous section we first determine the domination number of 3,5 and 6,5 as a special cases then we determined the domination numbers of    Proof: Let the vertices of this graph labeled by: ( ,5 ) = {1,2,3, … , 5 } see Fig.10. Similarly from above theorems we consider for ≥ 4 and ≠ 6 the four following cases:   If ∈ implies that either in 1 or in 2 .
For ∈ 1 the proof is similarly in the case (1)