Total coalitions in graphs

Abstract

We define a total coalition in a graph G as a pair of disjoint subsets A ₁,A ₂ ⊆ A that satisfy the following conditions: (a) neither A ₁ nor A ₂ constitutes a total dominating set of G, and (b) A ₁ ∪ A ₂ constitutes a total dominating set of G. A total coalition partition of a graph G is a partition ϒ = {A ₁,A ₂, . . . ,A_k } of its vertex set such that no subset of ϒ acts as a total dominating set of G, but for every set A_i ∈ ϒ, there exists a set A_j ∈ ϒ such that A_i and A_j combine to form a total coalition. We define the total coalition number of G as the maximum cardinality of a total coalition partition of G, and we denote it by C_t (G). The purpose of this paper is to begin an investigation into the characteristics of total coalition in graphs.