Total coalitions in graphs

Abstract

We define a total coalition in a graph G as a pair of disjoint subsets A₁,A₂ ⊆ A that satisfy the following conditions: (a) neither A₁ nor A₂ constitutes a total dominating set of G, and (b) A₁ ∪ A₂ constitutes a total dominating set of G. A total coalition partition of a graph G is a partition ϒ = {A₁,A₂, . . . ,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.

Mathematics Subject Classification (2020):

• 05C69

Key words:

• Total dominating set
• coalition
• total coalition
• tree