The substantial independence number for the strong product of two graphs

Abstract

Given a graph G, a substantial independent set S is a non empty subset of the vertex set V of the graph G = (V, E) if (i) S is an independent set of G and (ii) every vertex in V\\S is adjacent to at most one vertex in S. The substantial independence number bS (G) is the maximum cardinality of a maximal substantial independent set. The strong product of two graphs is a graph with V(G.H) = V(G).V(H) and ((g1, h1) (g2, h2)) ϵ E(G.H) if one of the following holds:g1 g2 ϵ E(G) and h1 h2 ϵ E(H)g1 = g2 and h1 h2 ϵ E(H)g1 g2 ϵ E(G) and h1 = h2In this paper we study the substantial independence number and some bounds for the strong product of two graphs namely Pm.Pn, Cm.Pn and Cm.Cn.