Restrained Italian domination in graphs

Abstract

For a graphG =(V(G),E(G)), an Italian dominating function (ID function)f:V(G) → {0,1,2} has the property that for every vertexv∈V(G) withf(v) = 0, eithervis adjacent to a vertex assigned 2 underforvis adjacent to least two vertices assigned 1 underf. The weight of an ID function is ∑v∈V(G)f(v). The Italian domination number is the minimum weight taken over all ID functions ofG. In this paper, we initiate the study of a variant of ID functions. A restrained Italian dominating function (RID function)fofGis an ID function ofGfor which the subgraph induced by {v ∈V(G) |f(v) = 0} has no isolated vertices, and the restrained Italian domination numberγrI(G) is the minimum weight taken over all RID functions ofG. We first prove that the problem of computing this parameter is NP-hard, even when restricted to bipartite graphs and chordal graphs as well as planar graphs with maximum degree five. We prove that γrI(T) for a treeTof ordern≥ 3 different from the double starS2,2can be bounded from below by (n+ 3)/2. Moreover, all extremal trees for this lower bound are characterized in this paper. We also give some sharp bounds on this parameter for general graphs and give the characterizations of graphsGwith small or largeγrI(G).