Trees with No Locating Roman Domination Critical Vertices

Abstract

A Roman dominating function (or just RDF) on a graph $$G =(V, E)$$ is a function $$f: V \longrightarrow \{0, 1, 2\}$$ 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 an RDF f is the value $$f(V(G))=\sum _{u \in V(G)}f(u)$$ . An RDF f can be represented as $$f=(V_0,V_1,V_2)$$ , where $$V_i=\{v\in V:f(v)=i\}$$ for $$i=0,1,2$$ . An RDF $$f=(V_0,V_1,V_2)$$ is called a locating Roman dominating function (or just LRDF) if $$N(u)\cap V_2\ne N(v)\cap V_2$$ for any pair u, v of distinct vertices of $$V_0$$ . The locating Roman domination number $$\gamma _R^L(G)$$ is the minimum weight of an LRDF of G. A vertex v of a graph G is called a locating Roman domination critical vertex (or just $$\gamma _R^L$$ critical vertex) if $$\gamma _R^L(G-v)<\gamma _R^L(G)$$ . In this paper, we characterize all trees with no $$\gamma _R^L$$ critical vertices.