Signed total double Roman dominatıon numbers in digraphs

Abstract

Let D = (V, A) be a finite simple digraph. A signed total double Roman dominating function (STDRD-function) on the digraph D is a function f : V (D) ? {?1, 1, 2, 3} satisfying the following conditions: (i) P x?N?(v) f(x) ? 1 for each v ? V (D), where N?(v) consist of all in-neighbors of v, and (ii) if f(v) = ?1, then the vertex v must have at least two in-neighbors assigned 2 under f or one in-neighbor assigned 3 under f, while if f(v) = 1, then the vertex v must have at least one in-neighbor assigned 2 or 3 under f. The weight of a STDRD-function f is the value P x?V (D) f(x). The signed total double Roman domination number (STDRD-number) ?tsdR(D) of a digraph D is the minimum weight of a STDRD-function on D. In this paper we study the STDRD-number of digraphs, and we present lower and upper bounds for ?tsdR(D) in terms of the order, maximum degree and chromatic number of a digraph. In addition, we determine the STDRD-number of some classes of digraphs.