Restrained Italian reinforcement number in graphs

A restrained Italian dominating function (RID-function) on a graph G=(V,E) is f:V→{0,1,2} satisfying: (i) f(N(u))≥2 for every vertex u∈V(G) with f(u)=0, where N(u) the set of vertices adjacent to u; (ii) subgraph induced by assigned 0 under f has no isolated vertices. The weight an RID-function sum its value over whole vertices, and domination number minimum G. In this paper, we initiate study reinforcement rrI(G) G defined as cardinality smallest edges that must add decrease number. We begin showing decision problem associated NP-hard arbitrary graphs. Then several properties well some sharp bounds are presented.