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Roman domination value in graphs | ||
| Communications in Combinatorics and Optimization | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 11 دی 1403 اصل مقاله (457.02 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2024.28899.1769 | ||
| نویسندگان | ||
| P. Roushini Leely Pushpam* 1؛ Padmapriea Sampath* 2 | ||
| 1Department of Mathematics, D.B. Jain College, Chennai 600 097, Tamil Nadu, India | ||
| 2Department of Mathematics, Sri Sairam Engineering College, Chennai 600 044, Tamil Nadu, India | ||
| چکیده | ||
| For a graph $G=(V,E)$, a set $S \subseteq V$ is a \textit{dominating set} if every vertex in $V\setminus S$ has a neighbour in $S$. The \textit{domination number}, denoted by $\gamma(G)$, is the minimum cardinality of a dominating set in $G$ and a dominating set of minimum cardinality is called a \textit{$\gamma(G)$-set}. Cockayne et al. defined a \textit{Roman dominating function} (RDF) on a graph $G = (V,E)$ to be a function $f:V\rightarrow \lbrace 0,1,2\rbrace$ satisfying the condition that every vertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex $v$ for which $f(v)=2$. The \textit{Roman domination number}, denoted by $\gamma_R(G)$, is the minimum weight of an RDF in $G$. An RDF of weight $\gamma_R(G)$ is called a \textit{$\gamma_R(G)$-function}. Eunjeong Yi introduced the \textit{domination value of $v$}, denoted by $DV_G(v)$, to be the number of $\gamma(G)$-sets to which $v$ belongs. In this paper, we extend the idea of domination value to Roman domination. For a vertex $v \in V$, we define the \textit{Roman domination value}, denoted by $R_G(v)$, as $ R_G(v) = \sum_{f \in \mathcal{F}} f(v)$, where $\mathcal{F}$ denote the set of all $\gamma_R(G)$-functions. We also study some basic properties of Roman domination value of vertices for a given graph and determine the Roman domination value for the vertices of a complete $k$-partite graph. | ||
| کلیدواژه‌ها | ||
| Domination؛ Roman domination؛ Roman domination value | ||
| مراجع | ||
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