Signed total double Roman dominatıon numbers in digraphs

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) γtsdR(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 γtsdR(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.Publisher's Versio

Signed total Roman domination in digraphs

Let D be a finite and simple digraph with vertex set V (D). A signed total Roman dominating function (STRDF) on a digraph D is a function f : V (D) → {−1, 1, 2} satisfying the conditions that (i) ∑x∈N−(v) f(x) ≥ 1 for each v ∈ V (D), where N−(v) consists of all vertices of D from which arcs go into v, and (ii) every vertex u for which f(u) = −1 has an inner neighbor v for which f(v) = 2. The weight of an STRDF f is w(f) = ∑v∈V (D) f(v). The signed total Roman domination number γstR(D) of D is the minimum weight of an STRDF on D. In this paper we initiate the study of the signed total Roman domination number of digraphs, and we present different bounds on γstR(D). In addition, we determine the signed total Roman domination number of some classes of digraphs. Some of our results are extensions of known properties of the signed total Roman domination number γstR(G) of graphs G

Signed Total Roman Domination in Digraphs

Let $D$ be a finite and simple digraph with vertex set $V (D)$. A signed total Roman dominating function (STRDF) on a digraph $D$ is a function $f : V (D) \rightarrow {−1, 1, 2}$ satisfying the conditions that (i) $\Sigma_{x \in N^− (v) } f(x) \ge 1$ for each $v \in V (D)$, where $N^− (v)$ consists of all vertices of $D$ from which arcs go into $v$, and (ii) every vertex u for which $f(u) = −1$ has an inner neighbor $v$ for which $f(v) = 2$. The weight of an STRDF $f$ is $w(f) = \Sigma_{ v \in V } (D) f(v)$. The signed total Roman domination number $\gamma_{stR} (D)$ of $D$ is the minimum weight of an STRDF on $D$. In this paper we initiate the study of the signed total Roman domination number of digraphs, and we present different bounds on $\gamma_{stR} (D)$. In addition, we determine the signed total Roman domination number of some classes of digraphs. Some of our results are extensions of known properties of the signed total Roman domination number $\gamma_{stR} (G)$ of graphs $G$