Connected Domination in Plane Triangulations
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929261397409792 |
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| author | Bryant, Felicity Pavelescu, Elena |
| author_facet | Bryant, Felicity Pavelescu, Elena |
| contents | A set of vertices of a graph $G$ such that each vertex of $G$ is either in the set or is adjacent to a vertex in the set is called a dominating set of $G$. If additionally, the set of vertices induces a connected subgraph of $G$ then the set is a connected dominating set of $G$. The domination number $γ(G)$ of $G$ is the smallest number of vertices in a dominating set of $G$, and the connected domination number $γ_c(G)$ of $G$ is the smallest number of vertices in a connected dominating set of $G$. We find the connected domination numbers for all triangulations of up to thirteen vertices. For $n\ge 15$, $n\equiv 0$ (mod 3), we find graphs of order $n$ and $γ_c=\frac{n}{3}$. We also show that the difference $γ_c(G)-γ(G)$ can be arbitrarily large. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_00595 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Connected Domination in Plane Triangulations Bryant, Felicity Pavelescu, Elena Combinatorics Geometric Topology 05C10, 57M15 A set of vertices of a graph $G$ such that each vertex of $G$ is either in the set or is adjacent to a vertex in the set is called a dominating set of $G$. If additionally, the set of vertices induces a connected subgraph of $G$ then the set is a connected dominating set of $G$. The domination number $γ(G)$ of $G$ is the smallest number of vertices in a dominating set of $G$, and the connected domination number $γ_c(G)$ of $G$ is the smallest number of vertices in a connected dominating set of $G$. We find the connected domination numbers for all triangulations of up to thirteen vertices. For $n\ge 15$, $n\equiv 0$ (mod 3), we find graphs of order $n$ and $γ_c=\frac{n}{3}$. We also show that the difference $γ_c(G)-γ(G)$ can be arbitrarily large. |
| title | Connected Domination in Plane Triangulations |
| topic | Combinatorics Geometric Topology 05C10, 57M15 |
| url | https://arxiv.org/abs/2403.00595 |