Main Article Content
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, . . . , Ak} of its vertex set such that no subset of Υ acts as a total dominating set of G, but for every set Ai ∈ Υ, there exists a set Aj ∈ Υ such that Ai and Aj 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 Ct(G). The purpose of this paper is to begin an investigation into the characteristics of total coalition in graphs.