On restrained coalitions in graphs: bounds and exact values
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| Format: | Preprint |
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2025
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| _version_ | 1866914197469659136 |
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| author | Dobrynin, Andrey A. Glebov, Aleksey N. Golmohammadi, H. |
| author_facet | Dobrynin, Andrey A. Glebov, Aleksey N. Golmohammadi, H. |
| contents | 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 $π$ such that every non-dominating set of $π$ is a member of some coalition, and every dominating set is a single-vertex set in $π$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_11440 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On restrained coalitions in graphs: bounds and exact values Dobrynin, Andrey A. Glebov, Aleksey N. Golmohammadi, H. Combinatorics 05C69 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 $π$ such that every non-dominating set of $π$ is a member of some coalition, and every dominating set is a single-vertex set in $π$. 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. |
| title | On restrained coalitions in graphs: bounds and exact values |
| topic | Combinatorics 05C69 |
| url | https://arxiv.org/abs/2512.11440 |