Complexity of Roman {2}-domination and the double Roman domination in graphs

Abstract

For a simple, undirected graph a Roman {2}-dominating function (R2DF) has the property that for every vertex with f(v) = 0, either there exists a vertex with f(u) = 2, or at least two vertices with The weight of an R2DF is the sum The minimum weight of an R2DF is called the Roman {2}-domination number and is denoted by A double Roman dominating function (DRDF) on G is a function such that for every vertex if f(v) = 0, then v has at least two neighbors with or one neighbor w with f(w) = 3, and if f(v) = 1, then v must have at least one neighbor w with The weight of a DRDF is the value The minimum weight of a DRDF is called the double Roman domination number and is denoted by Given an graph G and a positive integer k, the R2DP (DRDP) problem is to check whether G has an R2DF (DRDF) of weight at most k. In this article, we first show that the R2DP problem is NP-complete for star convex bipartite graphs, comb convex bipartite graphs and bisplit graphs. We also show that the DRDP problem is NP-complete for star convex bipartite graphs and comb convex bipartite graphs. Next, we show that are obtained in linear time for bounded tree-width graphs, chain graphs and threshold graphs, a subclass of split graphs. Finally, we propose a -approximation algorithm for the minimum Roman {2}-domination problem and -approximation algorithm for the minimum double Roman domination problem, where Δ is the maximum degree of G.