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Quasi total double Roman domination in trees | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 13، دوره 9، شماره 1، خرداد 2024، صفحه 159-168 اصل مقاله (414.69 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2023.29008.1809 | ||
| نویسندگان | ||
| Maryam Akhoundi* 1؛ Aysha Khan2؛ Jana Shafi3؛ Lutz Volkmann4 | ||
| 1Clinical Research Development Unit of Rouhani Hospital, Babol University of Medical Sciences, Babol 4717647745, Iran | ||
| 2Department of Mathematics, Prince Sattam bin Abdulaziz University, Alkharj 11991, Saudi Arabia | ||
| 3Department of Computer Science, College of Arts and Science, Prince Sattam bin Abdul Aziz University, Wadi Ad-Dwasir 11991, Saudi Arabia | ||
| 4RWTH Aachen University, 52056 Aachen, Germany | ||
| چکیده | ||
| A quasi total double Roman dominating function (QTDRD-function) on a graph $G=(V(G),E(G))$ is a function $f:V(G)\longrightarrow\{0,1,2,3\}$ having the property that \textrm{(i)} if $f(v)=0$, then vertex $v$ must have at least two neighbors assigned 2 under $f$ or one neighbor $w$ with $f(w)=3$; \textrm{(ii)} if $f(v)=1$, then vertex $v$ has at least one neighbor $w$ with $f(w)\geq2$, and \textrm{(iii)} if $x$ is an isolated vertex in the subgraph induced by the set of vertices assigned non-zero values, then $f(x)=2$. The weight of a QTDRD-function $f$ is the sum of its function values over the whole vertices, and the quasi total double Roman domination number $\gamma_{qtdR}(G)$ equals the minimum weight of a QTDRD-function on $G$. In this paper, we show that for any tree $T$ of order $n\ge 4$, $\gamma_{qtdR}(T)\le n+\frac{s(T)}{2}$, where $s(T)$ is the number of support vertices of $T$, that improves a known bound. | ||
| کلیدواژه‌ها | ||
| quasi total double Roman domination؛ total double Roman domination؛ double Roman domination number؛ Roman domination number | ||
| مراجع | ||
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