On restrained coalitions in graphs: bounds and exact values

TL;DR

This paper introduces the concept of restrained dominating coalitions in graphs, establishes bounds relating to coalition numbers, and determines these numbers specifically for cycles and trees.

Contribution

It defines restrained dominating coalitions and proves that their maximum number is bounded above by the coalition number, also computing these values for specific graph classes.

Findings

Proved that $RC(G) \\le C(G)$ for any graph $G$.

Determined the restrained coalition numbers for cycles.

Calculated the restrained coalition numbers for trees.

Abstract

A subset $D \subseteq V$ is a dominating set of a graph $G$ with vertex set $V$ if every vertex $v \in V \setminus D$ is adjacent to a vertex in $D$. Two subsets of $V$ form a coalition if neither of them is a dominating set, but their union is a dominating set. A coalition partition of $G$ is its vertex partition $\pi$ such that every non-dominating set of $\pi$ is a member of some coalition, and every dominating set is a single-vertex set in $\pi$. The coalition number $C(G)$ of a graph $G$ is the maximum cardinality of its coalition partitions. A subset $R \subseteq V$ is a restrained dominating set if $R$ is a dominating set and any vertex of $V \setminus R$ has at least one neighbor in $V \setminus R$. Restrained dominating coalition, restrained dominating partition and restrained coalition number $RC(G)$ are defined by the same way. In this paper, we prove that $RC(G) \le C(G)$ for an arbitrary graph $G$. In addition, the restrained coalition numbers of cycles and trees are determined.