On algorithmic complexity of double Roman domination

Abstract

A double Roman dominating function (DRDF) on a graph G=(V,E) is a function f:V⟶{0,1,2,3} such that every vertex v∈V with f(v)=0 is either adjacent to a vertex u with f(u)=3 or two distinct vertices x and y with f(x)=f(y)=2, and every vertex v∈V with f(v)=1 is adjacent to a vertex u with f(u)≥2. The weight of f is the sum f(V)=∑v∈Vf(v). The minimum weight of a DRDF on G is the double Roman domination number of G, denoted by γdR(G). A graph G is a double Roman Graph if γdR(G)=3γ(G), where γ(G) is the domination number of G.In this paper, we first show that the decision problem associated to double Roman domination is NP-complete even when restricted to planar graphs. Then, we study the complexity issue of a problem posed in [R.A. Beeler, T.W. Haynesa and S.T. Hedetniemi, Double Roman domination, Discrete Appl. Math. 211 (2016), 23–29], and show that the problem of deciding whether a given graph is double Roman is NP-hard even when restricted to bipartite or chordal graphs. Then, we give linear algorithms that compute the domination number and the double Roman domination number of a given unicyclic graph. Finally, we give a linear algorithm that decides whether a given unicyclic graph is double Roman.