Strong subgraph $k$-connectivity bounds

Let $D=(V,A)$ be a digraph of order $n$, $S$ a subset of $V$ of size $k$ and $2\le k\le n$. 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 $A(D_i)\cap A(D_j)=\emptyset$forall$1\le i<j\le p$\mathrm{.}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 $${\kappa }_k(D)=\min \{{\kappa }_S(D)\mid S\subseteq V,\mathrm{\mid }S\mathrm{\mid }=k\}\mathrm{.}$$ A digraph $D=(V,A)$ is called minimally strong subgraph $(k,\mathrm{\ell })$-connected if ${\kappa }_k(D)\ge \mathrm{\ell }$ 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 $(k,\mathrm{\ell })$-connected digraphs.