Corporate Domination Number of the Cartesian Product of Cycle and Path

Abstract

Domination in graphs is to dominate the graph G by a set of vertices <img src=image/13421349_01.gif>, vertex set of G) when each vertex in G is either in D or adjoining to a vertex in D. D is called a perfect dominating set if for each vertex v is not in D, which is adjacent to exactly one vertex of D. We consider the subset C which consists of both vertices and edges. Let <img src=image/13421349_02.gif> denote the set of all vertices V and the edges E of the graph G. Then <img src=image/13421349_03.gif> is said to be a corporate dominating set if every vertex v not in <img src=image/13421349_04.gif> is adjacent to exactly one vertex of <img src=image/13421349_04.gif>, where the set P consists of all vertices in the vertex set of an edge induced sub graph <img src=image/13421349_05.gif>, (E₁ a subset of E) such that there should be maximum one vertex common to any two open neighborhood of different vertices in V(G[E₁]) and Q, the set consists of all vertices in the vertex set V₁, a subset of V such that there exists no vertex common to any two open neighborhood of different vertices in V₁. The corporate domination number of G, denoted by <img src=image/13421349_06.gif>, is the minimum cardinality of elements in C. In this paper, we intend to determine the exact value of corporate domination number for the Cartesian product of the Cycle <img src=image/13421349_07.gif> and Path <img src=image/13421349_08.gif>.