Abstract
For a simple, undirected graph [Formula: see text], a restrained Roman dominating function (rRDF) [Formula: see text] has the property that, every vertex [Formula: see text] with [Formula: see text] is adjacent to at least one vertex v for which [Formula: see text] and at least one vertex [Formula: see text] for which [Formula: see text]. The weight of an rRDF is the sum [Formula: see text]. The minimum weight of an rRDF is called the restrained Roman domination number (rRDN) and is denoted by [Formula: see text]. 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.
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