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

For a simple, undirected graph G=(V,E), a Roman {2}-dominating function (R2DF) f:V -> {0,1,2} has the property that for every vertex v is an element of V with f(v) = 0, either there exists a vertex u is an element of N(v), with f(u) = 2, or at least two vertices x,y is an element of N(v) with f(x)=f(y)=1. The weight of an R2DF is the sum f(V)=<mml:munder>Sigma v is an element of V</mml:munder>f(v). The minimum weight of an R2DF is called the Roman {2}-domination number and is denoted by gamma {R2}(G). A double Roman dominating function (DRDF) on G is a function f:V -> {0,1,2,3} such that for every vertex v is an element of V if f(v) = 0, then v has at least two neighbors x,y is an element of N(v) with f(x)=f(y)=2 or one neighbor w with f(w) = 3, and if f(v) = 1, then v must have at least one neighbor w with f(w)>= 2. The weight of a DRDF is the value f(V)=<mml:munder>Sigma v is an element of V</mml:munder>f(v). The minimum weight of a DRDF is called the double Roman domination number and is denoted by gamma dR(G). 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 gamma {R2}(G),and gamma dR(G) are obtained in linear time for bounded tree-width graphs, chain graphs and threshold graphs, a subclass of split graphs. Finally, we propose a 2(1+ln(Delta +1))-approximation algorithm for the minimum Roman {2}-domination problem and 3(1+ln(Delta +1))-approximation algorithm for the minimum double Roman domination problem, where Delta is the maximum degree of G.