Double Roman domination in graphs: algorithmic complexity

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, for each v ∈ V with f(v) = 0, there is a vertex u adjacent to v with f(u) = 3 or there are vertices x and y adjacent to v such that f(x) = f(y) = 2 and for each v ∈ V with f(v) = 1, there is a vertex u adjacent to v with f(u) 1. The weight of a DRDF f is f(V ) = ∑ v∈V f(v). Let n and k be integers such that 3 ≤ 2k + 1 ≤ n. The generalized Petersen graph GP(n, k) = (V, E) is the graph with V = {u1, u2, . . . , un} ∪ {v1, v2, . . . , vn} and E = {uiui+1, uivi, vivi+k : 1 ≤ i ≤ n}, where addition is taken modulo n. In this paper, we firstly prove that the decision problem associated with double Roman domination is NP-complete even restricted to planar bipartite graphs with maximum degree at most 4. Next, we give a dynamic programming algorithm for computing a minimum DRDF (i.e., a DRDF with minimum weight along all DRDFs) of GP(n, k) in O(n81k) time and space and so a minimum DRDF of GP(n, O(1)) can be computed in O(n) time and space.