Internally-disjoint directed pendant Steiner trees with three terminal vertices in Cartesian product digraphs
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910022291685376 |
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| author | Yu, Shanshan Sun, Yuefang |
| author_facet | Yu, Shanshan Sun, Yuefang |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_13781 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Internally-disjoint directed pendant Steiner trees with three terminal vertices in Cartesian product digraphs Yu, Shanshan Sun, Yuefang Combinatorics Discrete Mathematics 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. |
| title | Internally-disjoint directed pendant Steiner trees with three terminal vertices in Cartesian product digraphs |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2602.13781 |