Internally-disjoint Directed Pendant Steiner Trees in Digraphs

TL;DR

This work defines and studies the intern-ally-disjoint directed pendant Steiner tree packing problem (IDPSTP) in digraphs, introducing the local parameter $\tau_{S,r}(D)$ and the global $\tau_{k}(D)$. It establishes a complexity dichotomy: on Eulerian digraphs, deciding whether $\tau_{S,r}(D)\ge\ell$ is NP-complete for fixed $|S|=k\ge3$ and $\ell\ge2$, while on connected symmetric digraphs the problem is polynomial-time solvable for fixed $k\ge3$, $\ell\ge2$ but NP-complete when $\ell$ or $k$ is part of the input. The paper also provides sharp, general bounds and extremal results for $\tau_k(D)$, including an arc-cut based upper bound and Nordhaus–Gaddum-type relations, thereby extending pendant-tree connectivity notions from undirected graphs to the directed setting and linking them to classical vertex-connectivity and directed Steiner-tree packing. These results clarify the algorithmic landscape and extremal behavior of directed pendant Steiner structures with potential implications for network design and reliability analysis in digraph models.

Abstract

For a digraph $D=(V(D),A(D))$ and a set $S\subseteq V(D)$ with $|S|\geq 2$ and $r\in S$, a directed pendant $(S,r)$-Steiner tree (or, simply, a pendant $(S,r)$-tree) is an out-tree $T$ rooted at $r$ such that $S\subseteq V(T)$ and each vertex of $S$ has degree one in $T$. Two pendant $(S,r)$-trees are called internally-disjoint if they are arc-disjoint and their common vertex set is exactly $S$. The goal of the {\sc Internally-disjoint Directed Pendant Steiner Tree Packing (IDPSTP)} problem is to find a largest collection of pairwise internally-disjoint pendant $(S,r)$-trees in $D$. We use $τ_{S,r}(D)$ to denote the maximum number of pairwise internally-disjoint pendant $(S,r)$-trees in $D$ and define the directed pendant-tree $k$-connectivity of $D$ as \begin{align*} τ_{k}(D)=\min\{τ_{S,r}(D)\mid S\subseteq V(D),|S|=k,r\in S\}. \end{align*} IDPSTP is a restriction of the {\sc Internally-disjoint Directed Steiner Tree Packing} problem studied by Cheriyan and Salavatipour [Algorithmica, 2006] and Sun and Yeo [JGT, 2023]. The directed pendant-tree $k$-connectivity extends the concept of pendant-tree $k$-connectivity in undirected graphs studied by Hager [JCTB, 1985] and could be seen as a generalization of classical vertex-connectivity of digraphs. In this paper, we completely determine the computational complexity for the parameter $τ_{S,r}(D)$ on Eulerian digraphs and symmetric digraphs. We also give sharp bounds and values for the parameter $τ_{k}(D)$.

Figures (4)

• Figure 1: Two $(S, r)$-trees with $S=\{r, v_1, v_2\}$
• Figure 2: The Eulerian digraph $D$
• Figure 3: The symmetric digraph $D$
• Figure 4: The symmetric digraph $D$

Theorems & Definitions (20)

• Theorem 2.1
• Theorem 2.2
• proof
• Lemma 2.1
• Theorem 2.3
• proof
• Lemma 2.2
• Theorem 2.4
• proof
• Theorem 2.5