Bounds on the Signed Roman k-Domination Number of a Digraph

Let $k$ be a positive integer. A signed Roman $k$-dominating function (SRkDF) 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 k $ for each $ v \in V (D) $, where $ N^− [v] $ is the closed in-neighborhood of $v$, and (ii) each vertex $u$ for which $f(u) = −1$ has an in-neighbor $v$ for which $f(v) = 2$. The weight of an SRkDF $f$ is $ \Sigma_{ v \in V (D) } f(v) $. The signed Roman $k$-domination number $ \gamma_{sR}^k (D) $ of a digraph $D$ is the minimum weight of an SRkDF on $D$. We determine the exact values of the signed Roman $k$-domination number of some special classes of digraphs and establish some bounds on the signed Roman $k$-domination number of general digraphs. In particular, for an oriented tree $T$ of order $n$, we show that $ \gamma_{sR}^2 (T) \ge (n + 3)//2 $, and we characterize the oriented trees achieving this lower bound.