Roman Domination Critical Graphs Upon Edge Subdivision

Nader Jafari Rad1
1Department of Mathematics, Shahrood University of Technology, Shahrood, Iran

Abstract

A function \( f: V(G) \to \{0, 1, 2\} \) is a \emph{Roman dominating function} (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 \emph{Roman domination number} 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.

Keywords: Roman domination; Subdivision, Critical. 2000 Mathematical subject classification: 05C69.