Strong subgraph $k$-connectivity bounds

Let $D=(V,A)$ be a digraph of order $n$, $S$ a subset of $V$ of size $k$ and $2\le k\leq n$. Strong subgraphs $D1, \dots , Dp$ containing $S$ are said to be internally disjoint if $V(Di)\cap V(Dj)=S$ and $A(Di)\cap A(Dj)=\emptyset$ for all $1\le i<j\le p$. Let $\kappaS(D)$ be the maximum number of internally disjoint strong digraphs containing $S$ in $D$. The strong subgraph $k$-connectivity is defined as $$\kappak(D)=\min{\kappaS(D)\mid S\subseteq V, |S|=k}.$$ A digraph $D=(V, A)$ is called minimally strong subgraph $(k,\ell)$-connected if $\kappak(D)\geq \ell$ but for any arc $e\in A$, $\kappak(D-e)\leq \ell-1$. In this paper, we first give a sharp upper bound for the parameter $\kappak(D)$ and then study the minimally strong subgraph $(k,\ell)$-connected digraphs.