Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 11 3 2026 09 01 Roman domination value in graphs 1033 1046 14880 10.22049/cco.2024.28899.1769 EN P. Roushini Leely Pushpam Department of Mathematics, D.B. Jain College, Chennai 600 097, Tamil Nadu, India Padmapriea Sampath Department of Mathematics, Sri Sairam Engineering College, Chennai 600 044, Tamil Nadu, India Journal Article 2023 08 03 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. https://comb-opt.azaruniv.ac.ir/article_14880_bbe95ffa92249d4b7a71766b84723aa1.pdf