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arXiv · 2404.11575

Strong coalitions in graphs

Hamidreza Golmohammadi·Saeid Alikhani·Nima Ghanbari·I. I. Takhonov·A. Abaturov
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Abstract

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 $\Omega=\{V_1, V_2,..., V_k \}$ of vertices in $G$ is a strong coalition partition, if every set $V_i \in\Omega$ 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 $V_j \in \Omega$ 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.

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Hamidreza Golmohammadi, Saeid Alikhani, Nima Ghanbari, I. I. Takhonov, A. Abaturov. 2024-04-17. Strong coalitions in graphs. https://arxiv.org/abs/2404.11575

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