On double Roman domination problem for several graph classes.

<italic>A double Roman domination function</italic> (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 mapping 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 the conditions: (<italic>i</italic>) each vertex with 0 assigned is adjacent to a vertex with 3 assigned or at least two vertices with 2 assigned and (<italic>ii</italic>) each vertex with 1 assigned is adjacent to at least one vertex with 2 or 3 assigned. The weight of a DRDF <italic>f</italic> is defined as the sum ∑v∈Vf(v)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sum _{v\in V}f(v)$$\end{document}. The minimum weight of a DRDF on a graph <italic>G</italic> is called the <italic>double Roman domination number</italic> (DRDN) of <italic>G</italic>. This study establishes the values on DRDN for several graph classes. The exact values of DRDN are proved for Kneser graphs Kn,k,n≥k(k+2)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$K_{n,k},n\ge k(k+2)$$\end{document}, Johnson graphs Jn,2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$J_{n,2}$$\end{document}, for a few classes of convex polytopes, and the flower snarks. Moreover, tight lower and upper bounds on SRDN are proved for some convex polytopes. For the generalized Petersen graphs Pn,3,n≢0(mod4)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$P_{n,3}, n \not \equiv 0\,(\mathrm {mod\ 4})$$\end{document}, we make a further improvement on the best known upper bound from the literature. [ABSTRACT FROM AUTHOR]