New bounds on the outer-independent total double Roman domination number

A double Roman dominating function (DRDF) on a graph G = (V,E) is a function f : V ->{0, 1, 2, 3} satisfying (i) if f(v) = 0 then there must be at least two neighbors assigned two under f or one neighbor w with f(w) = 3; and (ii) if f(v) = 1 then v must be adjacent to a vertex w such that f(w) =2. 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 domination number ?(oi)(tdR)(G) is the minimum weight of an OITDRDF on G. In this paper, we establish various bounds on ?(oi)(tdR)(G). In particular, we present Nordhaus-Gaddum-type inequalities for this parameter. Some of our results improve the previous results.