Let $D=(V(D),A(D))$ be a digraph with a terminal vertex subset $S\subseteq V(D)$ such that $|S|=k\geq 2$. An out-tree $T$ of $D$ rooted at $r$ is called a directed pendant $(S,r)$-Steiner tree (or, pendant $(S,r)$-tree for short) if $r\in S\subseteq V(T)$ and $d_{T}^{+}(r)=d_{T}^{-}(u)=1$ for each $u\in S\backslash \{r\}$. Two pendant $(S,r)$-trees $T_{1}$ and $T_{2}$ are internally-disjoint if $A(T_{1})\cap A(T_{2})=\varnothing$ and $V(T_{1})\cap V(T_{2})=S$. The pendant-tree $k$-connectivity $τ_{k}(D)$ of $D$ is defined as $$τ_{k}(D)=\min\{τ_{S,r}(D)\mid S\subseteq V(D),|S|=k,r\in S\},$$ where $τ_{S,r}(D)$ denotes the maximum number of pairwise internally-disjoint pendant $(S,r)$-trees in $D$. In this paper, we derive a sharp lower bound for the pendant-tree 3-connectivity of the Cartesian product digraph $D\square H$, where $D$ and $H$ are both strong digraphs. Specifically, we prove the lower bound $τ_{3}(D\square H)\geq τ_{3}(D)+τ_{3}(H)$. Moreover, we propose a polynomial-time algorithm for finding internally-disjoint pendant $(S,r)$-trees which attain this lower bound.

翻译：令 $D=(V(D),A(D))$ 为一个有向图，其终端顶点子集 $S\subseteq V(D)$ 满足 $|S|=k\geq 2$。若 $D$ 中一棵以 $r$ 为根的出树 $T$ 满足 $r\in S\subseteq V(T)$ 且对每个 $u\in S\backslash \{r\}$ 有 $d_{T}^{+}(r)=d_{T}^{-}(u)=1$，则称 $T$ 为一棵有向悬挂 $(S,r)$-Steiner树（简称为悬挂 $(S,r)$-树）。两棵悬挂 $(S,r)$-树 $T_{1}$ 和 $T_{2}$ 称为内部不相交的，若 $A(T_{1})\cap A(T_{2})=\varnothing$ 且 $V(T_{1})\cap V(T_{2})=S$。$D$ 的悬挂树 $k$-连通度 $τ_{k}(D)$ 定义为 $$τ_{k}(D)=\min\{τ_{S,r}(D)\mid S\subseteq V(D),|S|=k,r\in S\},$$ 其中 $τ_{S,r}(D)$ 表示 $D$ 中两两内部不相交的悬挂 $(S,r)$-树的最大数目。本文中，我们推导了笛卡尔积有向图 $D\square H$ 的悬挂树 3-连通度的一个尖锐下界，其中 $D$ 和 $H$ 均为强连通有向图。具体而言，我们证明了该下界 $τ_{3}(D\square H)\geq τ_{3}(D)+τ_{3}(H)$。此外，我们提出了一种多项式时间算法，用于构造达到此下界的内部不相交悬挂 $(S,r)$-树。