Introduction To Coalitions In Graphs

A coalition in a graphG = ( V , E )consists of two disjoint sets of vertices V-1 and V-2, neither of which is a dominating set but whose unionV1V2is a dominating set. A coalition partition in a graph G of ordern = | V |is a vertex partitionpi = {V1,V2, ... ,Vk}such that every set V-i of pi either is a dominating set consisting of a single vertex of degree n - 1, or is not a dominating set but forms a coalition with another setVjis an element of piwhich is not a dominating set. In this paper we introduce this concept and study its properties.