Restrained Global Defensive Alliances in Graphs

Abstract

A defensive alliance in a graph G is a nonempty set of vertices S ⊆ V (G) such that for every vertex v ∈ S, |N[v] ∩ S| ≥ |N(v) ∩ (V (G) ∖ S)|. A defensive alliance S is called global if every vertex in V (G) ∖ S is adjacent to at least one member of the alliance S. In this paper, the concept of restrained global defensive alliance in graphs was introduced. In particular, a global defensive alliance S is a restrained global defensive alliance if the induced subgraph of V ∖ S has no isolated vertex. Here, some properties of this alliance were identified, and its bounds were also determined. In addition, the restrained global defensive alliance number was also formulated, along with characterizations of some special classes of graphs, specifically complete, complete bipartite, and path graphs.