Trees with No Locating Roman Domination Critical Vertices

Nader Jafari Rad | Ali Taherifar | Hadi Rahbani

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URL :   http://research.shahed.ac.ir/WSR/WebPages/Report/PaperView.aspx?PaperID=158314
Date :  2020/11/21
Publish in :    Iranian Journal of Science and Technology, Transactions A: Science

Link :  https://link.springer.com/article/10.1007/s40995-020-01023-x
Keywords :Roman dominating function

Abstract :

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A Roman dominating function (or just RDF) on a graph G=(V,E) is a function f:V⟶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))=∑u∈V(G)f(u). An RDF f can be represented as f=(V0,V1,V2), where Vi=v∈V:f(v)=i for i=0,1,2. An RDF f=(V0,V1,V2) is called a locating Roman dominating function (or just LRDF) if N(u)∩V2≠N(v)∩V2 for any pair u, v of distinct vertices of V0. The locating Roman domination number γLR(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 γLR critical vertex) if γLR(G−v)γLR(G). In this paper, we characterize all trees with no γLR critical vertices.