Roman Domination Critical Graphs Upon Edge Subdivision

Abstract

A function $f:V(G)\to \{0,1,2\}$ is a $Romandominatingfunction$ (or just RDF) if every vertex $u$ for which $f(u)=0$ is adjacent to at least one vertex $v$ for which $f(v)=2$. The weight of a Roman dominating function is the value $f(V(G))=\sum _{u\in V(G)}f(u)$. The $Romandominationnumber$ of a graph $G$, denoted by ${\gamma }_R(G)$, is the minimum weight of a Roman dominating function on $G$. A graph $G$ is Roman domination critical upon edge subdivision if the Roman domination number increases whenever an edge is subdivided. In this paper, we study the Roman domination critical graphs upon edge subdivision. We present several properties, bounds, and general results for these graphs.