Research Article

Total coalitions in graphs

DOI: 10.2989/16073606.2024.2365365
Author(s): Saeid Alikhani Yazd University, Iran, Davood Bakhshesh University of Bojnord, Iran, Hamidreza Golmohammadi Novosibirsk State University, Russia,

Abstract

We define a total coalition in a graph G as a pair of disjoint subsets A 1,A 2A that satisfy the following conditions: (a) neither A 1 nor A 2 constitutes a total dominating set of G, and (b) A 1A 2 constitutes a total dominating set of G. A total coalition partition of a graph G is a partition ϒ = {A 1,A 2, . . . ,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.

Get new issue alerts for Quaestiones Mathematicae