ALGORITHMIC ASPECTS OF ROMAN GRAPHS

Let G = (V, E) be a graph. A set S ⊆ V is called a dominating set of G if for every v ∈ V \S there is at least one vertex u ∈ N(v) such that u ∈ S. The domination number of G, denoted by γ(G), is equal to the minimum cardinality of a dominating set in G. A Roman dominating function (RDF) on G is a function f : V → {0, 1, 2} such that every vertex v ∈ V with f(v) = 0 is adjacent to at least one vertex u with f(u) = 2. The weight of f is the sum f(V ) = ∑ v∈V f(v). The minimum weight of a RDF on G is the Roman domination number of G, denoted by γR(G). A graph G is a Roman Graph if γR(G) = 2γ(G). In this paper, we first study the complexity issue of the problem posed in [E. J. Cockayane, P. M. Dreyer Jr., S. M. Hedetniemi and S. T. Hedetniemi, On Roman domination in graphs, Discrete Math. 278 (2004), 11–22], and show that the problem of deciding whether a given graph is a Roman graph is NP-hard even when restricted to chordal graphs. Then, we give linear algorithms that compute the domination number and the Roman domination number of a given unicyclic graph. Finally, using these algorithms we give a linear algorithm that decides whether a given unicyclic graph is a Roman graph.