Algorithm and hardness results in double Roman domination of graphs

Abstract

Let G=(V,E) be a graph. A double Roman dominating function (DRDF) of G is a function f:V→{0,1,2,3} such that (i) each vertex v with f(v)=0 is adjacent to either a vertex u with f(u)=3 or two vertices u1 and u2 with f(u1)=f(u2)=2, and (ii) each vertex v with f(v)=1 is adjacent to a vertex u with f(u)>1. The double Roman domination number of G is the minimum weight of a DRDF along all DRDFs on G, where the weight of a DRDF f on G is f(V)=∑v∈Vf(v). In this paper, we first propose an algorithm to compute the double Roman domination number of an interval graph G=(V,E) in O(|V|+|E|) time, answering a problem posed in Banerjee et al. (2020) [2]. Next, we show that the decision problem associated with the double Roman domination is NP-complete for split graphs. Finally, we show that the computational complexities of the Roman domination problem and the double Roman domination problem are different.