Extremal Results for Directed Tree Connectivity

Abstract

For a digraph $D=(V(D), A(D))$, and a set $S\subseteq V(D)$ with $r\in S$ and $|S|\ge 2$, an (S, r)-tree is an out-tree T rooted at r with $S\subseteq V(T)$. 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 internally disjoint and arc-disjoint (S, r)-trees in D, respectively. The generalized k-vertex-strong connectivity of D is defined as

$$\begin{aligned} \kappa _k(D)= \min \{\kappa _{S,r}(D)\mid S\subseteq V(D), |S|=k, r\in S\}. \end{aligned}$$

Similarly, the generalized k-arc-strong connectivity of D is defined as

$$\begin{aligned} \lambda _k(D)= \min \{\lambda _{S,r}(D)\mid S\subseteq V(D), |S|=k, r\in S\}. \end{aligned}$$

The generalized k-vertex-strong connectivity and generalized k-arc-strong connectivity are also called directed tree connectivity which could be seen as a generalization of classical connectivity of digraphs and a natural extension of undirected tree connectivity. A digraph $D=(V(D), A(D))$ is called minimally generalized $(k, \ell )$-vertex (respectively, arc)-strongly connected if $\kappa _k(D)\ge \ell$ (respectively, $\lambda _k(D)\ge \ell$) but for any arc $e\in A(D)$, $\kappa _k(D-e)\le \ell -1$ (respectively, $\lambda _k(D-e)\le \ell -1$). In this paper, we study the minimally generalized $(k, \ell )$-vertex (respectively, arc)-strongly connected digraphs. We compute the minimum and maximum sizes of these digraphs and give characterizations of such digraphs for some pairs of k and $\ell$.