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A new upper bound on the independent $2$-rainbow domination number in trees | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 20، دوره 8، شماره 1، خرداد 2023، صفحه 261-270 اصل مقاله (378.78 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2022.27641.1305 | ||
| نویسندگان | ||
| Nader Jafari Rad* 1؛ Elham Gholami2؛ A Tehranian3؛ Hamid Rasouli4 | ||
| 1Shahed University | ||
| 2Islamic Azad University | ||
| 3‎Islamic Azad University | ||
| 4Islamic Azad University, | ||
| چکیده | ||
| A $2$-rainbow dominating function on a graph $G$ is a function $g$ that assigns to each vertex a set of colors chosen from the subsets of $\{1, 2\}$ so that for each vertex with $g(v) =\emptyset$ we have $\bigcup_{u\in N(v)} g(u) = \{1, 2\}$. The weight of a $2$-rainbow dominating function $g$ is the value $w(g) = \sum_{v\in V(G)} |f(v)|$. A $2$-rainbow dominating function $g$ is an independent $2$-rainbow dominating function if no pair of vertices assigned nonempty sets are adjacent. The $2$-rainbow domination number $\gamma_{r2}(G)$ (respectively, the independent $2$-rainbow domination number $i_{r2}(G)$) is the minimum weight of a $2$-rainbow dominating function (respectively, independent $2$-rainbow dominating function) on $G$. We prove that for any tree $T$ of order $n\geq 3$, with $\ell$ leaves and $s$ support vertices, $i_{r2}(T)\leq (14n+\ell+s)/20$, thus improving the bound given in [Independent 2-rainbow domination in trees, Asian-Eur. J. Math. 8 (2015) 1550035] under certain conditions. | ||
| کلیدواژه‌ها | ||
| Rainbow domination؛ Independent rainbow domination؛ Tree | ||
| مراجع | ||
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