Algorithmic complexity of triple Roman dominating functions on graphs

Given a graph G = (V;E), a function f : V ! f0; 1; 2; 3; 4g is a triple Roman dominating function (TRDF) of G, for each vertex v 2 V , (i) if f(v) = 0, then v must have either one neighbour in V4, or either two neighbours in V2 [ V3 (one neighbour in V3) or either three neighbours in V2, (ii) if f(v) = 1, then v must have either one neighbour in V3 [ V4 or either two neighbours in V2, and if f(v) = 2, then v must have one neighbour in V2 [ V3 [ V4. The triple Roman domination number of G is the minimum weight of an TRDF f of G, where the weight of f is ∑ 𝑣(v). The triple Roman domination problem is to compute the triple Roman domination number of a given graph. In this paper, we study the triple Roman domination problem. We show that the problem is NP-complete for the star convex bipartite and the comb convex bipartite graphs and is APX-complete for graphs of degree at most 4. We propose a linear-time algorithm for computing the triple Roman domination number of proper interval graphs. We also give an (2H(Δ(G) + 1) - 1)-approximation algorithm for solving the problem for any graph G, where Δ(G) is the maximum degree of G and H(d) denotes the first d terms of the harmonic series. In addition, we prove that for any 0 there is no (1/4 ) ln jV j-approximation polynomial-time algorithm for solving the problem on bipartite and split graphs, unless NP DTIME (jV jO(log log jV j)).