RESTRAINED ROMAN REINFORCEMENT NUMBER IN GRAPHS

A restrained Roman dominating function (RRD-function) on a graph \(G=(V,E)\) is \(f\) from \(V\) into \(\{0,1,2\}\) satisfying: (i) every vertex \(u\) with \(f(u)=0\) adjacent to \(v\) \(f(v)=2\); (ii) the subgraph induced by vertices assigned 0 under has no isolated vertices. The weight of an RRD-function sum its value over whole set vertices, and domination number minimum \(G.\) In this paper, we begin study reinforcement \(r_{rR}(G)\) \(G\) defined as cardinality smallest edges that must add decrease number. We first show decision problem associated NP-hard. Then several properties well some sharp bounds are presented. particular it established \(r_{rR}(T)=1\) for tree \(T\) order at least three.