Document Type : Original Manuscript


Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran.



Let $G=(V, E)$ be a graph.
A set $S \subseteq V$ is called a dominating set of $G$ if for every $v\in V-S$ there is at least one vertex $u \in N(v)$ such that $u\in S$. 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 $f:V\longrightarrow\{0,1,2\}$ such that every vertex $v\in V$ with $f(v)=0$ is adjacent to at least one vertex $u$ with $f(u)=2$. The weight of $f$ is the sum $f(V)=\sum_{v\in V}f (v)$. 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

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.


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