ALGORITHMIC ASPECTS OF ROMAN GRAPHS

Let $G=(V, E)$ be a graph. A set $S subseteq V$ is called a dominating set of $G$ if for every $vin V-S$ there is at least one vertex $u in N(v)$ such that $uin S$. The domination number of $G$, denoted by $gamma(G)$, is equal to the minimum cardinality of a dominating set in $G$. A Roman dominating function (RDF) on $G$ is a function $f:Vlongrightarrow{0,1,2}$ such that every vertex $vin V$ with $f(v)=0$ is adjacent to at least one vertex $u$ with $f(u)=2$. The weight of $f$ is the sum $f(V)=sum_{vin V}f (v)$. The minimum weight of a RDF on $G$ is the Roman domination number of $G$, denoted by $gamma_R(G)$. A graph $G$ is a Roman Graph if $gamma_R(G)=2gamma(G)$. 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, textit{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.