Directed Steiner path packing and directed path connectivity

For a digraph $D=(V(D), A(D))$, and a set $S\subseteq V(D)$ with $r\in S$ and $|S|\geq 2$, a directed $(S, r)$-Steiner path or, simply, an $(S, r)$-path is a directed path $P$ started at $r$ with $S\subseteq V(P)$. Two $(S, r)$-paths are said to be arc-disjoint if they have no common arc. Two arc-disjoint $(S, r)$-paths are said to be internally disjoint if the set of common vertices of them is exactly $S$. Let $\kappa^p_{S,r}(D)$ (resp. $\lambda^p_{S,r}(D)$) be the maximum number of internally disjoint (resp. arc-disjoint) $(S, r)$-paths in $D$. The directed path $k$-connectivity of $D$ is defined as $$\kappa^p_k(D)= \min \{\kappa^p_{S,r}(D)\mid S\subseteq V(D), |S|=k, r\in S\}.$$ Similarly, the directed path $k$-arc-connectivity of $D$ is defined as $$\lambda^p_k(D)= \min \{\lambda^p_{S,r}(D)\mid S\subseteq V(D), |S|=k, r\in S\}.$$ The directed path $k$-connectivity and directed path $k$-arc-connectivity are also called directed path connectivity which extends the path connectivity on undirected graphs to directed graphs and could be seen as a generalization of classical connectivity of digraphs. In this paper, we obtain complexity results for $\kappa^p_{S,r}(D)$ on Eulerian digraphs and symmetric digraphs, and $\lambda^p_{S,r}(D)$ on general digraphs. We also give bounds for the parameters $\kappa^p_k(D)$ and $\lambda^p_k(D)$.