Signed total double Roman dominating functions in graphs

A signed total double Roman dominating function (STDRDF) on an isolated-free graph G= (V, E) is a function f: V (G) →−1, 1, 2, 3 such that (i) every vertex v with f (v) =−1 has at least two neighbors assigned 2 under f or one neighbor w with f (w) = 3, (ii) every vertex v with f (v) = 1 has at least one neighbor w with f (w) ≥2 and (iii) ∑u∈N (v) f (u) ≥1 holds for any vertex v. The weight of a STDRDF is the value f (V (G) ) =∑u∈V (G) f (u). The signed total double Roman domination number γsdRt (G) is the minimum weight of a STDRDF on G. In this article, we provide various bounds on γsdRt (G) and we show that the corresponding decision problem is NP-complete for bipartite and chordal graphs. In addition, we determine the signed total double Roman domination number of some classes of graphs including cycles, complete graphs and complete bipartite graphs.