An improved upper bound for the domination number of a graph
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arXiv
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| Format: | Preprint |
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2024
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| author | Arumugam, Subramanian Hegde, Suresh Manjanath Kulamarva, Shashanka |
| author_facet | Arumugam, Subramanian Hegde, Suresh Manjanath Kulamarva, Shashanka |
| contents | Let $G$ be a graph of order $n$. A classical upper bound for the domination number of a graph $G$ having no isolated vertices is $\lfloor\frac{n}{2}\rfloor$. However, for several families of graphs, we have $γ(G) \le \lfloor\sqrt{n}\rfloor$ which gives a substantially improved upper bound. In this paper, we give a condition necessary for a graph $G$ to have $γ(G) \le \lfloor\sqrt{n}\rfloor$, and some conditions sufficient for a graph $G$ to have $γ(G) \le \lfloor\sqrt{n}\rfloor$. We also present a characterization of all connected graphs $G$ of order $n$ with $γ(G) = \lfloor\sqrt{n}\rfloor$. Further, we prove that for a graph $G$ not satisfying $rad(G)=diam(G)=rad(\overline{G})=diam(\overline{G})=2$, deciding whether $γ(G) \le \lfloor\sqrt{n}\rfloor$ or $γ(\overline{G}) \le \lfloor\sqrt{n}\rfloor$ can be done in polynomial time. We conjecture that this decision problem can be solved in polynomial time for any graph $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_02765 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An improved upper bound for the domination number of a graph Arumugam, Subramanian Hegde, Suresh Manjanath Kulamarva, Shashanka Combinatorics Discrete Mathematics 05C69 Let $G$ be a graph of order $n$. A classical upper bound for the domination number of a graph $G$ having no isolated vertices is $\lfloor\frac{n}{2}\rfloor$. However, for several families of graphs, we have $γ(G) \le \lfloor\sqrt{n}\rfloor$ which gives a substantially improved upper bound. In this paper, we give a condition necessary for a graph $G$ to have $γ(G) \le \lfloor\sqrt{n}\rfloor$, and some conditions sufficient for a graph $G$ to have $γ(G) \le \lfloor\sqrt{n}\rfloor$. We also present a characterization of all connected graphs $G$ of order $n$ with $γ(G) = \lfloor\sqrt{n}\rfloor$. Further, we prove that for a graph $G$ not satisfying $rad(G)=diam(G)=rad(\overline{G})=diam(\overline{G})=2$, deciding whether $γ(G) \le \lfloor\sqrt{n}\rfloor$ or $γ(\overline{G}) \le \lfloor\sqrt{n}\rfloor$ can be done in polynomial time. We conjecture that this decision problem can be solved in polynomial time for any graph $G$. |
| title | An improved upper bound for the domination number of a graph |
| topic | Combinatorics Discrete Mathematics 05C69 |
| url | https://arxiv.org/abs/2401.02765 |