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

The Signed Roman Domatic Number of a Digraph

Let $D$ be a finite and simple digraph with vertex set $V(D)$.A {\em signed Roman dominating function} on the digraph $D$ isa function $f:V (D)\longrightarrow \{-1, 1, 2\}$ such that$\sum_{u\in N^-[v]}f(u)\ge 1$ for every $v\in V(D)$, where $N^-[v]$ consists of $v$ andall inner neighbors of $v$, and every vertex $u\in V(D)$ for which $f(u)=-1$ has an innerneighbor $v$ for which $f(v)=2$. A set $\{f_1,f_2,\ldots,f_d\}$ of distinct signedRoman dominating functions on $D$ with the property that $\sum_{i=1}^df_i(v)\le 1$ for each$v\in V(D)$, is called a {\em signed Roman dominating family} (of functions) on $D$. The maximumnumber of functions in a signed Roman dominating family on $D$ is the {\em signed Roman domaticnumber} of $D$, denoted by $d_{sR}(D)$. In this paper we initiate the study of signed Romandomatic number in digraphs and we present some sharp bounds for $d_{sR}(D)$. In addition, wedetermine the signed Roman domatic number of some digraphs. Some of our results are extensionsof well-known properties of the signed Roman domatic number of graphs