Neighborhood alliance in join of a graph with K1

Abstract

Let G = (V,E) be a graph. A subset S of V is called a neighborhood set of G if union of induced subgraph of N[s] is isomorphic to G, where union is taken over all s in S. A defensive alliance is a non-empty subset S of V satisfying the condition that every v ∈ S has at most one more neighbor in V − S than it has in S. The minimum cardinality of any defensive alliance of G is called the alliance number of G. Further, a subset of V which is both a neighborhood set of G as well as a defensive alliance of G is called a neighborhood alliance set, or simply an na-set. The minimum cardinality of an na-set is called neighborhood alliance number of G. The minimum cardinality (in possible cases) of various types of na-sets of join of a graph G with K₁, specifically when G is K_n−1, K_n, C_n and P_n are determined in this article.