Internally-disjoint directed pendant Steiner trees with three terminal vertices in Cartesian product digraphs

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 $\tau_{k}(D)$ of $D$ is defined as $$\tau_{k}(D)=\min\{\tau_{S,r}(D)\mid S\subseteq V(D),|S|=k,r\in S\},$$ where $\tau_{S,r}(D)$ 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 $\tau_{3}(D\square H)\geq \tau_{3}(D)+\tau_{3}(H)$. Moreover, we propose a polynomial-time algorithm for finding internally-disjoint pendant $(S,r)$-trees which attain this lower bound.