On the Outer Independent Total Double Roman Domination in Graphs

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 2 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 dominating number ?(oi) (tdR)(G) is the minimum weight of an OITDRDF on G. First, we show that the problem of determining ?(oi) (tdR)(G) 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 ?(oi) (tdR)(G) as well as the exact values for several graph families.