Algorithmic results in Roman dominating functions on graphs

Abstract

Given a graph G=(V,E), a subset D⊆V (respectively, function f:V→{0,1,2}) is a dominating set (DS) (respectively, Roman dominating function (RDF)) of G if each vertex v∈V∖D (respectively, v∈V with f(v)=0) is adjacent to a vertex u∈D (respectively, u∈V with f(u)=2). The domination number of G is the minimum cardinality of an DS of G and the Roman domination number of G is the minimum weight of an RDF f of G, where the weight of f is ∑v∈Vf(v). The (Roman) domination problem is to compute the (Roman) domination number of a given graph. In this paper, we study the Roman domination problem. We show that the complexity of the problem differs from the complexity of the domination problem and the problem is NP-complete for circle graphs and undirected path graphs and is APX-complete for graphs of degree at most 4. We also propose an integer linear programming (ILP) formulation with polynomial number of constraints for the problem. Additionally, we use the ILP formulation to give an H(Δ(G)+1)-approximation algorithm for solving the problem for any graph G, where Δ(G) is the maximum degree of G. Furthermore, we show that the optimization version of the problem on split and chordal graphs cannot be approximated in polynomial time within (1/2−ε)ln⁡|V| for any ε>0, unless NP ⊆ DTIME (|V|O(log⁡log⁡|V|)).