Roman Domination Dot-critical Graphs

A Roman dominating function on a graph G is a function f : V(G) → {0, 1, 2} 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 weight of a Roman dominating function is the value \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f(V(G))=\sum_{u \in V(G)}f(u)}$$\end{document}. The Roman domination number, γR(G), of G is the minimum weight of a Roman dominating function on G. In this paper, we study graphs for which contracting any edge decreases the Roman domination number.