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arXiv · 2512.11440

On restrained coalitions in graphs: bounds and exact values

Andrey A. Dobrynin·Aleksey N. Glebov·H. Golmohammadi
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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.

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Andrey A. Dobrynin, Aleksey N. Glebov, H. Golmohammadi. 2025-12-12. On restrained coalitions in graphs: bounds and exact values. https://arxiv.org/abs/2512.11440

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