Domination in 3rd dimensional product of vertex measurable graphs
Mehul Kumar, Saad Salman Ahmed, R. Murali · Journal of Information and Optimization Sciences · 2018
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 G1, G2, G3 denoted by G1 × G2 × G3 is a graph with vertex set . Such that two vertices a = (u1, v1, w1) and b = (u2, v2, w2) are said to be adjacent if u1 = u2 where v1 v2 ϵ and w1 w2 ϵ or v1 = v2 where u1 u2 ϵ and w1 w2 ϵ or w1 = w2 where u1 u2 ϵ and v1 v2 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 G1, G2, G3 having two vertices