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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2407.21472 |
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| _version_ | 1866910548498579456 |
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| author | Golmohammadi, Hamidreza |
| author_facet | Golmohammadi, Hamidreza |
| contents | Let $G(V, E)$ be a finite, simple, isolate-free graph. A set $D$ of vertices of a graph $G$ with the vertex set $V$ is a double dominating set of $G$, if every vertex $v\in D$ has at least one neighbor in $D$ and every vertex $v \in V \setminus D$ has at least two neighbors in $D$. A double coalition consists of two disjoint sets of vertices $V_{1}$ and $V_{2}$, neither of which is a double dominating set but their union $V_{1}\cup V_{2}$ is a double dominating set. A double coalition partition of a graph $G$ is a partition $Π= \{V_1, V_2,..., V_k \}$ of $V$ such that no subset of $Π$ is a double dominating set of $G$, but for every set $V_i \in Π$, there exists a set $V_j \in Π$ such that $V_i$ and $V_j$ form a double coalition. In this paper, we study properties of double coalitions in graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_21472 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Introduction to double coalitions in graphs Golmohammadi, Hamidreza Combinatorics Let $G(V, E)$ be a finite, simple, isolate-free graph. A set $D$ of vertices of a graph $G$ with the vertex set $V$ is a double dominating set of $G$, if every vertex $v\in D$ has at least one neighbor in $D$ and every vertex $v \in V \setminus D$ has at least two neighbors in $D$. A double coalition consists of two disjoint sets of vertices $V_{1}$ and $V_{2}$, neither of which is a double dominating set but their union $V_{1}\cup V_{2}$ is a double dominating set. A double coalition partition of a graph $G$ is a partition $Π= \{V_1, V_2,..., V_k \}$ of $V$ such that no subset of $Π$ is a double dominating set of $G$, but for every set $V_i \in Π$, there exists a set $V_j \in Π$ such that $V_i$ and $V_j$ form a double coalition. In this paper, we study properties of double coalitions in graphs. |
| title | Introduction to double coalitions in graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2407.21472 |