Strong in-domatic number in digraphs

Let $D=(V,A)$ be a digraph and $\mathfrak{S}$ a partition of $V(D)$. We say that $\mathfrak{S}$ is a strong in-domatic partition if every $S$ in $\mathfrak{S}$ holds that every vertex not in $S$ has at least one out-neighbor in $S$, that is $S$ is an in-dominating set, and $D\langle S \rangle$ is strongly connected. The maximum number of elements in a strong in-domatic partition is called the strong in-domatic number of $D$ and it is denoted by $\mathsf{d}_{s}^{-}(D)$. In this paper we introduce those concepts and determine the value of $\mathsf{d}_{s}^{-}$ for semicomplete digraphs and planar digraphs. We show some structural properties of digraphs which have a strong in-domatic partition and we see some bounds for $\mathsf{d}_{s}^{-}(D)$. Then we study this concept in the Cartesian product, composition, line digraph and other associated digraphs. In addition, we characterize strong in-domatic critical digraphs and we give two families strong in-domatic critical digraphs which hold some properties, where a strong in-domatic critical digraph $D$ holds that $\mathsf{d}_{s}^{-}(D-e) = \mathsf{d}_{s}^{-}(D) -1$ for every $e$ in $A(D)$.