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| Autori principali: | , , , , |
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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2410.06486 |
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| _version_ | 1866916428919078912 |
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| author | Gao, Hong Qiu, Daoda Du, Shuyan Zhao, Yiyue Yang, Yuansheng |
| author_facet | Gao, Hong Qiu, Daoda Du, Shuyan Zhao, Yiyue Yang, Yuansheng |
| contents | Given a graph $G$ with vertex set $V$, an outer independent Roman dominating function (OIRDF) is a function $f$ from $V(G)$ to $\{0, 1, 2\}$ for which every vertex with label $0$ under $f$ is adjacent to at least a vertex with label $2$ but not adjacent to another vertex with label $0$. The weight of an OIRDF $f$ is the sum of vertex function values all over the graph, and the minimum of an OIRDF is the outer independent Roman domination number of $G$, denoted as $γ_{oiR}(G)$. In this paper, we focus on the outer independent Roman domination number of the Cartesian product of paths and cycles $P_{n}\Box C_{m}$. We determine the exact values of $γ_{oiR}(P_n\Box C_m)$ for $n=1,2,3$ and $γ_{oiR}(P_n\Box C_3)$ and present an upper bound of $γ_{oiR}(P_n\Box C_m)$ for $n\ge 4, m\ge 4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_06486 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Outer Independent Roman Domination Number of Cartesian Product of Paths and Cycles Gao, Hong Qiu, Daoda Du, Shuyan Zhao, Yiyue Yang, Yuansheng Combinatorics Given a graph $G$ with vertex set $V$, an outer independent Roman dominating function (OIRDF) is a function $f$ from $V(G)$ to $\{0, 1, 2\}$ for which every vertex with label $0$ under $f$ is adjacent to at least a vertex with label $2$ but not adjacent to another vertex with label $0$. The weight of an OIRDF $f$ is the sum of vertex function values all over the graph, and the minimum of an OIRDF is the outer independent Roman domination number of $G$, denoted as $γ_{oiR}(G)$. In this paper, we focus on the outer independent Roman domination number of the Cartesian product of paths and cycles $P_{n}\Box C_{m}$. We determine the exact values of $γ_{oiR}(P_n\Box C_m)$ for $n=1,2,3$ and $γ_{oiR}(P_n\Box C_3)$ and present an upper bound of $γ_{oiR}(P_n\Box C_m)$ for $n\ge 4, m\ge 4$. |
| title | Outer Independent Roman Domination Number of Cartesian Product of Paths and Cycles |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2410.06486 |