Triple Roman domination subdivision number in graphs

• Published in: Computer science journal of Moldova Vol. 30; no. 1(88); pp. 109 - 130
• Main Authors: Amjadi, J., Sadeghi, H.
• Format: Journal Article
• Language: English
• Published: Vladimir Andrunachievici Institute of Mathematics and Computer Science 01.02.2022
• Subjects: triple roman domination number; triple roman domination subdivision number
• ISSN: 1561-4042; 2587-4330
• DOI: 10.56415/csjm.v30.07

For a graph $G=(V, E)$, a triple Roman domination function is a function $f: V(G)\longrightarrow\{0, 1, 2, 3, 4\}$ having the property that for any vertex $v\in V(G)$, if $f(v)<3$, then $f(\mbox{AN}[v])\geq|\mbox{AN}(v)|+3$, where $\mbox{AN}(v)=\{w\in N(v)\mid f(w)\geq1\}$ and $\mbox{AN}[v]=\mbox{AN}(v)\cup\{v\}$. The weight of a triple Roman dominating function $f$ is the value $\omega(f)=\sum_{v\in V(G)}f(v)$. The triple Roman domination number of $G$, denoted by $\gamma_{[3R]}(G)$, equals the minimum weight of a triple Roman dominating function on $G$. {\em The triple Roman domination subdivision number} $\mbox{sd}_{\gamma_{[3R]}}(G)$ of a graph $G$ is the minimum number of edges that must be subdivided (each edge in $G$ can be subdivided at most once) in order to increase the triple Roman domination number. In this paper, we first show that the decision problem associated with $\mbox{sd}_{\gamma_{[3R]}}(G)$ is NP-hard and then establish upper bounds on the triple Roman domination subdivision number for arbitrary graphs.