Strong coalitions in graphs
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909267136610304 |
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| author | Golmohammadi, Hamidreza Alikhani, Saeid Ghanbari, Nima Takhonov, I. I. Abaturov, A. |
| author_facet | Golmohammadi, Hamidreza Alikhani, Saeid Ghanbari, Nima Takhonov, I. I. Abaturov, A. |
| contents | For a graph $G=(V,E)$, a set $D\subset V(G)$ is a strong dominating set of $G$, if for every vertex $x\in V (G)\setminus D$ there is a vertex $y\in D$ with $xy \in E(G)$ and $deg(x)\leq deg(y)$. A strong coalition consists of two disjoint sets of vertices $V_{1}$ and $V_{2}$, neither of which is a strong dominating set but whose union $V_{1}\cup V_{2}$, is a strong dominating set. A vertex partition $Ω=\{V_1, V_2,..., V_k \}$ of vertices in $G$ is a strong coalition partition, if every set $V_i \inΩ$ either is a strong dominating set consisting of a single vertex of degree $n-1$, or is not a strong dominating set but produces a strong coalition with another set $V_j \in Ω$ that is not a strong dominating set. The maximum cardinality of a strong coalition partition of $G$ is the strong coalition number of $G$ and is denoted by $SC(G)$. In this paper, we study properties of strong coalitions in graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_11575 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Strong coalitions in graphs Golmohammadi, Hamidreza Alikhani, Saeid Ghanbari, Nima Takhonov, I. I. Abaturov, A. Combinatorics 05C60 For a graph $G=(V,E)$, a set $D\subset V(G)$ is a strong dominating set of $G$, if for every vertex $x\in V (G)\setminus D$ there is a vertex $y\in D$ with $xy \in E(G)$ and $deg(x)\leq deg(y)$. A strong coalition consists of two disjoint sets of vertices $V_{1}$ and $V_{2}$, neither of which is a strong dominating set but whose union $V_{1}\cup V_{2}$, is a strong dominating set. A vertex partition $Ω=\{V_1, V_2,..., V_k \}$ of vertices in $G$ is a strong coalition partition, if every set $V_i \inΩ$ either is a strong dominating set consisting of a single vertex of degree $n-1$, or is not a strong dominating set but produces a strong coalition with another set $V_j \in Ω$ that is not a strong dominating set. The maximum cardinality of a strong coalition partition of $G$ is the strong coalition number of $G$ and is denoted by $SC(G)$. In this paper, we study properties of strong coalitions in graphs. |
| title | Strong coalitions in graphs |
| topic | Combinatorics 05C60 |
| url | https://arxiv.org/abs/2404.11575 |