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For a digraph $D=(V(D), A(D))$, and a set $S\subseteq V(D)$ with $r\in S$ and$|S|\geq 2$, an $(S, r)$-tree is an out-tree $T$ rooted at $r$ with $S\subseteqV(T)$. Two $(S, r)$-trees $T_1$ and $T_2$ are said to be arc-disjoint if$A(T_1)\cap A(T_2)=\emptyset$. Two arc-disjoint $(S, r)$-trees $T_1$ and $T_2$are said to be internally disjoint if $V(T_1)\cap V(T_2)=S$. Let$\kappa_{S,r}(D)$ and $\lambda_{S,r}(D)$ be the maximum number of internallydisjoint and arc-disjoint $(S, r)$-trees in $D$, respectively. The generalized$k$-vertex-strong connectivity of $D$ is defined as $$\kappa_k(D)= \min\{\kappa_{S,r}(D)\mid S\subset V(D), |S|=k, r\in S\}.$$ Similarly, thegeneralized $k$-arc-strong connectivity of $D$ is defined as $$\lambda_k(D)=\min \{\lambda_{S,r}(D)\mid S\subset V(D), |S|=k, r\in S\}.$$ The generalized$k$-vertex-strong connectivity and generalized $k$-arc-strong connectivity arealso called directed tree connectivity which extends the well-established treeconnectivity on undirected graphs to directed graphs and could be seen as ageneralization of classical connectivity of digraphs.  In this paper, we completely determine the complexity for both $\kappa_{S,r}(D)$ and $\lambda_{S, r}(D)$ on general digraphs, symmetric digraphs andEulerian digraphs. In particular, among our results, we prove and use theNP-completeness of 2-linkage problem restricted to Eulerian digraphs. We alsogive sharp bounds and characterizations for the two parameters $\kappa_k(D)$and $\lambda_k(D)$.

Directed Steiner tree packing and directed tree connectivity