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 $D_1, \dots , D_p$ containing $S$ are said to be internally disjoint if $V(D_i)\cap V(D_j)=S$ and $A(D_i)\cap A(D_j)=\emptyset$ for all $1\le i<j\le p$. Let $\kappa_S(D)$ be the maximum number of internally disjoint strong digraphs containing $S$ in $D$. The strong subgraph $k$-connectivity is defined as $$\kappa_k(D)=\min\{\kappa_S(D)\mid S\subseteq V, |S|=k\}.$$ A digraph $D=(V, A)$ is called minimally strong subgraph $(k,\ell)$-connected if $\kappa_k(D)\geq \ell$ but for any arc $e\in A$, $\kappa_k(D-e)\leq \ell-1$. In this paper, we first give a sharp upper bound for the parameter $\kappa_k(D)$ and then study the minimally strong subgraph $(k,\ell)$-connected digraphs.