A New Approach on Roman Graphs (original) (raw)

2021

Abstract

Let G=(V,E)G=(V,E)G=(V,E) be a simple graph with vertex set V=V(G)V=V(G)V=V(G) and edge set E=E(G)E=E(G)E=E(G). A Roman dominating function (RDF) on a graph GGG is a function f:Vrightarrow0,1,2f:V\rightarrow\{0,1,2\}f:Vrightarrow0,1,2 satisfying the condition that every vertex uuu for which f(u)=0f(u)=0f(u)=0 is adjacent to at least one vertex vvv such that f(v)=2f(v)=2f(v)=2. The weight of fff is omega(f)=SigmavinVf(v)\omega(f)=\Sigma_{v\in V}f(v)omega(f)=SigmavinVf(v). The minimum weight of an RDF on GGG, gammaR(G)\gamma_{R}(G)gammaR(G), is called the Roman domination number of GGG. gammaR(G)leq2gamma(G)\gamma_{R}(G)\leq 2\gamma(G)gammaR(G)leq2gamma(G) where gamma(G)\gamma(G)gamma(G) denotes the domination number of GGG. A graph GGG is called a Roman graph whenever gammaR(G)=2gamma(G)\gamma_{R}(G)= 2\gamma(G)gammaR(G)=2gamma(G). On the other hand, the differential of XXX is defined as partial(X)=∣B(X)∣−∣X∣\partial(X)=|B(X)|-|X|partial(X)=∣B(X)∣−∣X∣ and the differential of a graph GGG, written partial(G)\partial(G)partial(G), is equal to maxpartial(X):XsubseteqVmax\{\partial(X): X\subseteq V\}maxpartial(X):XsubseteqV. By using differential we provide a sufficient and necessary condition for the graphs to be Roman. We also modify the proof of a result on Roman trees. Finally we characterize the large family of tre...

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