Signed double Roman domination numbers in digraphs

Abstract

Let $D=(V,A)$ be a finite simple digraph. A signed double Roman dominating function (SDRD-function) on the
digraph $D$ is a function $f:V(D)\rightarrow\{-1,1,2, 3\}$ satisfying the following conditions:
(i) $\sum_{x\in N^-[v]}f(x)\ge 1$ for each $v\in V(D)$, where $N^-[v]$ consist of $v$ and 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, while if $f(v)=1$, then the vertex $v$ must have at least one in-neighbor assigned 2 or 3. The weight of a SDRD-function $f$ is the value $\sum_{x\in V(D)}f(x)$. The signed double Roman domination number (SDRD-number) $\gamma_{sdR}(D)$ of a digraph $D$ is the minimum weight of a SDRD-function on $D$. In this paper we study the SDRD-number of digraphs, and we present lower and upper bounds for $\gamma_{sdR}(D)$ in terms of the order, maximum degree and chromatic number of a digraph. In addition, we determine the SDRD-number of some classes of digraphs.