Domination in 3rd dimensional product of vertex measurable graphs. Issue 8 (17th November 2018) - Kumar, M. Kamal; Ahmed, Saad Salman; Murali, R.
Abstract: Consider a simple connected graph G ( V, E ), a set S is a dominating set if for every vertex u ϵ V – S, there exists a vertex v ϵ S such that u is adjacent to v . i.e. for every vertex u ϵ V – S, d ( u, S ) = 1. A dominating set D in G is a minimal dominating set if no proper subset of D is a dominating set. The minimum cardinality among all the minimal dominating sets is called domination number of the graph G denoted by g ( G ). Let be three simple non trivial connected graph. The 3rd dimension product of vertex measurable graphs of G 1, G 2, G 3 denoted by G 1 × G 2 × G 3 is a graph with vertex set . Such that two vertices a = ( u 1, v 1, w 1 ) and b = ( u 2, v 2, w 2 ) are said to be adjacent if u 1 = u 2 where v 1 v 2 ϵ and w 1 w 2 ϵ or v 1 = v 2 where u 1 u 2 ϵ and w 1 w 2 ϵ or w 1 = w 2 where u 1 u 2 ϵ and v 1 v 2 ϵ . In this paper we introduce few extended definitions of 3rd dimensional product of vertex measurable graphs, apply the concept of domination sets for 3rd dimension product of vertex measurable graphs with G 1, G 2, G 3 having two vertices