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On the outer independent 2-rainbow domination number of Cartesian products of paths and cycles | ||
| Communications in Combinatorics and Optimization | ||
| دوره 6، شماره 2، اسفند 2021، صفحه 315-324 اصل مقاله (351.73 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2021.27067.1188 | ||
| نویسنده | ||
| Nasrin Dehgardi* | ||
| Sirjan University of Technology, Sirjan 78137, Iran | ||
| چکیده | ||
| Let $G$ be a graph. A $2$-rainbow dominating function (or {\em 2-RDF}) of $G$ is a function $f$ from $V(G)$ to the set of all subsets of the set $\{1,2\}$ such that for a vertex $v\in V(G)$ with $f(v)=\emptyset$, the condition $\bigcup_{u\in N_{G}(v)}f(u)=\{1,2\}$ is fulfilled, where $N_{G}(v)$ is the open neighborhood of $v$. The weight of 2-RDF $f$ of $G$ is the value $\omega (f):=\sum _{v\in V(G)}|f(v)|$. The {\em $2$-rainbow domination number} of $G$, denoted by $\gamma_{r2}(G)$, is the minimum weight of a 2-RDF of $G$. A 2-RDF $f$ is called an outer independent $2$-rainbow dominating function (or OI2-RDF} of $G$ if the set of all $v\in V(G)$ with $f(v)=\emptyset$ is an independent set. The outer independent $2$-rainbow domination number $\gamma_{oir2}(G)$ is the minimum weight of an OI2-RDF of $G$. In this paper, we obtain the outer independent $2$-rainbow domination number of $P_{m}\square P_{n}$ and $P_{m}\square C_{n}$. Also we determine the value of $\gamma_{oir2}(C_{m}\Box C_{n})$ when $m$ or $n$ is even. | ||
| کلیدواژهها | ||
| 2-rainbow dominating function؛ 2-rainbow domination number؛ outer independent 2-rainbow dominating function؛ outer independent 2-rainbow domination number؛ Cartesian product | ||
| مراجع | ||
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