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arXiv · 2410.06486

Outer Independent Roman Domination Number of Cartesian Product of Paths and Cycles

Hong Gao·Daoda Qiu·Shuyan Du·Yiyue Zhao·Yuansheng Yang
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Abstract

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 $\gamma_{oiR}(G)$. 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 $\gamma_{oiR}(P_n\Box C_m)$ for $n=1,2,3$ and $\gamma_{oiR}(P_n\Box C_3)$ and present an upper bound of $\gamma_{oiR}(P_n\Box C_m)$ for $n\ge 4, m\ge 4$.

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Hong Gao, Daoda Qiu, Shuyan Du, Yiyue Zhao, Yuansheng Yang. 2024-10-09. Outer Independent Roman Domination Number of Cartesian Product of Paths and Cycles. https://arxiv.org/abs/2410.06486

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