Domination and packing in graphs
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916118150512640 |
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| author | Gómez, Renzo Gutiérrez, Juan |
| author_facet | Gómez, Renzo Gutiérrez, Juan |
| contents | Given a graph~$G$, the domination number, denoted by~$γ(G)$, is the minimum cardinality of a dominating set in~$G$. Dual to the notion of domination number is the packing number of a graph. A packing of~$G$ is a set of vertices whose pairwise distance is at least three. The packing number~$ρ(G)$ of~$G$ is the maximum cardinality of one such set. Furthermore, the inequality~$ρ(G) \leq γ(G)$ is well-known. Henning et al.\ conjectured that~$γ(G) \leq 2ρ(G)+1$ if~$G$ is subcubic. In this paper, we progress towards this conjecture by showing that~${γ(G) \leq \frac{120}{49}ρ(G)}$ if~$G$ is a bipartite cubic graph. We also show that $γ(G) \leq 3ρ(G)$ if~$G$ is a maximal outerplanar graph, and that~$γ(G) \leq 2ρ(G)$ if~$G$ is a biconvex graph. Moreover, in the last case, we show that this upper bound is tight. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_05088 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Domination and packing in graphs Gómez, Renzo Gutiérrez, Juan Combinatorics Discrete Mathematics 05C69 G.2.2 Given a graph~$G$, the domination number, denoted by~$γ(G)$, is the minimum cardinality of a dominating set in~$G$. Dual to the notion of domination number is the packing number of a graph. A packing of~$G$ is a set of vertices whose pairwise distance is at least three. The packing number~$ρ(G)$ of~$G$ is the maximum cardinality of one such set. Furthermore, the inequality~$ρ(G) \leq γ(G)$ is well-known. Henning et al.\ conjectured that~$γ(G) \leq 2ρ(G)+1$ if~$G$ is subcubic. In this paper, we progress towards this conjecture by showing that~${γ(G) \leq \frac{120}{49}ρ(G)}$ if~$G$ is a bipartite cubic graph. We also show that $γ(G) \leq 3ρ(G)$ if~$G$ is a maximal outerplanar graph, and that~$γ(G) \leq 2ρ(G)$ if~$G$ is a biconvex graph. Moreover, in the last case, we show that this upper bound is tight. |
| title | Domination and packing in graphs |
| topic | Combinatorics Discrete Mathematics 05C69 G.2.2 |
| url | https://arxiv.org/abs/2402.05088 |