SIGNED TOTAL DOUBLE ROMAN DOMINATION NUMBERS IN DIGRAPHS

SIGNED TOTAL DOUBLE ROMAN DOMINATION NUMBERS IN DIGRAPHS

J. Amjadi, F. P. Hosseini

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Abstract

Let D = (V;A) be a nite simple digraph. A signed total double Roman dominating function (STDRD-function) on the digraph D is a function f : V (D) ! f-1; 1; 2; 3g satisfying the following conditions: (i) P x2N-(v) f(x)  1 for each v 2 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 as- signed 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 x2V (D) f(x). The signed total double Roman domination number (STDRD-number) t sdR(D) of a di- graph 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 t sdR(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.

Keywords

signed total double Roman dominating function, signed total double Roman domination number, directed graph