A function f:V(G)→{0,1,2}\documentclass[12pt]{minimal}
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\begin{document}$$f:V(G)\rightarrow \{0,1,2\}$$\end{document} is a Roman dominating function (RDF) if every vertex u for which f(u)=0\documentclass[12pt]{minimal}
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\begin{document}$$f(u)=0$$\end{document} is adjacent to at least one vertex v for which f(v)=2\documentclass[12pt]{minimal}
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\begin{document}$$f(v)=2$$\end{document}. The weight of a Roman dominating function is the value f(V(G))=∑u∈Vf(u)\documentclass[12pt]{minimal}
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\begin{document}$$f(V(G))=\sum _{u \in V}f(u)$$\end{document}. The Roman domination number of a graph G, denoted by γR(G)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _{R}(G)$$\end{document}, is the minimum weight of a Roman dominating function on G. A connected (respectively, total) Roman dominating function is an RDF f such that the vertices with non-zero labels under f induce a connected graph (respectively, a subgraph with no isolated vertex). The connected (respectively, total) Roman domination number of a graph G, denoted by γcR(G)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _{cR}(G)$$\end{document} (respectively, γtR(G)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _{tR}(G)$$\end{document}) is the minimum weight of a connected (respectively, total) RDF of G. It this paper we first study the complexity issue of the problems posed in [H. Abdollahzadeh Ahangar, M. A. Henning, V. Samodivkin and I. G. Yero, Total Roman domination in graphs, Appl. Anal. Discret. Math. 10 (2016), 501–517], and show that the problem of deciding whether γtR(G)=2γ(G)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _{tR}(G)=2\gamma (G)$$\end{document}, γtR(G)=2γt(G)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _{tR}(G)=2\gamma _t(G)$$\end{document} or γtR(G)=3γ(G)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _{tR}(G)=3\gamma (G)$$\end{document} is NP-hard even when restricted to chordal or bipartite graphs. Then, we give a linear algorithm that decides whether γtR(G)=2γ(G)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _{tR}(G)=2\gamma (G)$$\end{document}, γtR(G)=2γt(G)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _{tR}(G)=2\gamma _t(G)$$\end{document} or γtR(G)=3γ(G)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _{tR}(G)=3\gamma (G)$$\end{document}, if G is a tree or a unicyclic graph.
