On the Outer Independent Total Double Roman Domination in Graphs

A double Roman dominating function (DRDF) on a graph G=(V,E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=(V,E)$$\end{document} is a function f:V→{0,1,2,3}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:V\rightarrow \{0,1,2,3\}$$\end{document} satisfying (i) if f(v)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(v)=0$$\end{document}, then there must be at least two neighbors assigned 2 under f or one neighbor w with f(w)=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(w)=3$$\end{document}; and (ii) if f(v)=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(v)=1$$\end{document} then v must be adjacent to a vertex w, such that f(w)≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(w)\ge 2$$\end{document}. A DRDF is an outer independent total double Roman dominating function (OITDRDF) on G if the set of vertices labeled 0 induces an edgeless subgraph and the subgraph induced by the vertices with a non-zero label has no isolated vertices. The weight of an OITDRDF is the sum of its function values over all vertices, and the outer independent total Roman dominating number γtdRoi(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _{tdR}^{oi}(G)$$\end{document} is the minimum weight of an OITDRDF on G. First, we show that the problem of determining γtdRoi(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _{tdR}^{oi}(G)$$\end{document} is NP-complete for bipartite and chordal graphs. Then, we show that it is solvable in linear time when we are restricting to bounded clique-width graphs. Moreover, we present some tight bounds on γtdRoi(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _{tdR}^{oi}(G)$$\end{document} as well as the exact values for several graph families.