Global coalition sets in graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911163250376704 |
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| author | Besharati, Nazli Mojdeh, Doost Ali Samadzadeh, Mohammad Reza |
| author_facet | Besharati, Nazli Mojdeh, Doost Ali Samadzadeh, Mohammad Reza |
| contents | Let $G=(V,E)$ be a graph. A subset $S \subseteq V$ is called a global dominating set of $G$, if it serves as a dominating set in both $G$ and its complement $\overline{G}$. We define two disjoint subsets $V_1,V_2 \subseteq V$ to form a global coalition if neither $V_1$ nor $V_2$ individually constitutes a global dominating set, yet their union $V_1 \cup V_2$ does. A global coalition partition (abbreviated as $gc$-partition) of $G$ is a vertex partition $π$ of $V(G)$ such that for every subset $V_i \in π$, there exists another subset $V_j \in π$ with which $V_i$ forms a global coalition. In this paper, we initiate the study of global coalition in graphs. Specifically, we prove that every graph admits a gc-partition. Additionally, we establish an upper bound on the number of global coalitions in which each member of a gc-partition can participate. We also explore the relationships between global coalition and coalition, as well as between global coalition and perfect coalition in graphs. Finally, we explore properties of $gc$-partitions in unicyclic graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_15386 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global coalition sets in graphs Besharati, Nazli Mojdeh, Doost Ali Samadzadeh, Mohammad Reza Combinatorics Let $G=(V,E)$ be a graph. A subset $S \subseteq V$ is called a global dominating set of $G$, if it serves as a dominating set in both $G$ and its complement $\overline{G}$. We define two disjoint subsets $V_1,V_2 \subseteq V$ to form a global coalition if neither $V_1$ nor $V_2$ individually constitutes a global dominating set, yet their union $V_1 \cup V_2$ does. A global coalition partition (abbreviated as $gc$-partition) of $G$ is a vertex partition $π$ of $V(G)$ such that for every subset $V_i \in π$, there exists another subset $V_j \in π$ with which $V_i$ forms a global coalition. In this paper, we initiate the study of global coalition in graphs. Specifically, we prove that every graph admits a gc-partition. Additionally, we establish an upper bound on the number of global coalitions in which each member of a gc-partition can participate. We also explore the relationships between global coalition and coalition, as well as between global coalition and perfect coalition in graphs. Finally, we explore properties of $gc$-partitions in unicyclic graphs. |
| title | Global coalition sets in graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.15386 |