For a given graph G without isolated vertex we consider 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}. For every i∈{0,1,2}\documentclass[12pt]{minimal}
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\begin{document}$$i\in \{0,1,2\}$$\end{document}, let Vi={v∈V(G):f(v)=i}\documentclass[12pt]{minimal}
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\begin{document}$$V_i=\{v\in V(G):\; f(v)=i\}$$\end{document}. The function f is known to be an outer-independent total Roman dominating function for the graph G if it is satisfied that; (i) every vertex in V0\documentclass[12pt]{minimal}
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\begin{document}$$V_0$$\end{document} is adjacent to at least one vertex in V2\documentclass[12pt]{minimal}
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\begin{document}$$V_2$$\end{document}; (ii) V0\documentclass[12pt]{minimal}
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\begin{document}$$V_0$$\end{document} is an independent set; and (iii) the subgraph induced by V1∪V2\documentclass[12pt]{minimal}
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\begin{document}$$V_1\cup V_2$$\end{document} has no isolated vertex. The minimum possible weight ω(f)=∑v∈V(G)f(v)\documentclass[12pt]{minimal}
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\begin{document}$$\omega (f)=\sum _{v\in V(G)}f(v)$$\end{document} among all outer-independent total Roman dominating functions for G is called the outer-independent total Roman domination number of G. In this article we obtain new tight bounds for this parameter that improve some well-known results. Such bounds can also be seen as relationships between this parameter and several other classical parameters in graph theory like the domination, total domination, Roman domination, independence, and vertex cover numbers. In addition, we compute the outer-independent total Roman domination number of Sierpiński graphs, circulant graphs, and the Cartesian and direct products of complete graphs.
