Quasi total double Roman domination in graphs

AbstractA quasi total double Roman dominating function (QTDRD-function) on a graph G=(V(G),E(G))is a function f:V(G)→{0,1,2,3}having the property that (i) if f(v) = 0, then vertex vmust have at least two neighbors assigned 2 under for one neighbor wwith f(w) = 3; (ii) if f(v) = 1, then vertex vhas at least one neighbor wwith f(w)≥2, and (iii) if xis an isolated vertex in the subgraph induced by the set of vertices assigned nonzero values, then f(x) = 2. The weight of a QTDRD-function fis the sum of its function values over the whole vertices, and the quasi total double Roman domination number γqtdR(G)equals the minimum weight of a QTDRD-function on G. In this paper, we first show that the problem of computing the quasi total double Roman domination number of a graph is NP-hard, and then we characterize graphs Gwith small or large γqtdR(G). Moreover, we establish an upper bound on the quasi total double Roman domination number and we characterize the connected graphs attaining this bound.