Total restrained coalitions in graphs

Abstract

A vertex set in a graph without isolated vertices is a total restrained dominating set (TRD-set) if it is dominating, induces a subgraph without isolated vertices, and the vertices not in the set also induce a subgraph without isolated vertices. Two vertex sets, which are not TRD-sets, form a total restrained coalition if their union is a TRD-set. A total restrained coalition partition is a partition where none of its elements are TRD-sets, but each forms a total restrained coalition with another element. The goal is to maximize the cardinality of such a partition, denoted C tr (G). We initiate the study of this concept by proving certain properties, extremal values, general bounds, and its relation to known structural parameters. Exact values for specific graph families are also provided.