Complexity aspects of restrained Roman domination in graphs.

For a simple, undirected graph G = (V , E) , a restrained Roman dominating function (rRDF) f : V → { 0 , 1 , 2 } has the property that, every vertex u with f (u) = 0 is adjacent to at least one vertex v for which f (v) = 2 and at least one vertex w for which f (w) = 0. The weight of an rRDF is the sum f (V) = ∑ v ∈ V f (v). The minimum weight of an rRDF is called the restrained Roman domination number (rRDN) and is denoted by γ r R (G). We show that restrained Roman domination and domination problems are not equivalent in computational complexity aspects. Next, we show that the problem of deciding if G has an rRDF of weight at most l for chordal and bipartite graphs is NP-complete. Finally, we show that rRDN is determined in linear time for bounded treewidth graphs and threshold graphs. [ABSTRACT FROM AUTHOR]