三次图与系统发育网络的无传递弧定向

中文摘要 AI 辅助

无向图$G$的$st$-定向是一种有向无环图，它通过为$G$的每条边指定方向得到，仅含一个源点$s$和一个汇点$t$。判断无向图$G$是否存在$st$-定向的经典问题可高效求解；另一方面，判断$G$是否存在不含任何传递弧的$st$-定向是NP完全问题，即便$G$的每个顶点度数至多为4。本文证明，若$G$是三次图，该判定问题仍为NP完全，这解决了Binucci等人（2025）提出的开放问题。我们针对该问题的两种变体得到了NP完全性：（i）$s$和$t$是固定的，作为输入的一部分给出；（ii）$s$和$t$可自由选择。随后，我们利用这些结果研究计算进化中出现的一个问题的计算复杂性，具体而言，我们证明判断无根二元系统发育网络是否可定向为不含任何捷径（系统发育中传递弧的类似物）的有根二元系统发育网络的问题是NP完全的。我们的结果将定向无向图和定向无根系统发育网络这两个（大多）不同的研究领域联系起来。

英文摘要

An $st$-orientation of an undirected graph $G$ is an acyclic digraph with a single source $s$ and a single sink $t$ that can be obtained from $G$ by assigning a direction to each edge. The classical problem of deciding if an undirected graph $G$ has an $st$-orientation can be solved efficiently. On the other hand, deciding if an $st$-orientation of $G$ exists that does not have any transitive arc is NP-complete, even if each vertex of $G$ has degree at most four. Here we show that this last decision problem remains NP-complete if $G$ is cubic, which settles an open question by Binucci et al. (2025). We obtain NP-completeness for two variants of the problem: (i) $s$ and $t$ are fixed and given as part of the input and (ii) $s$ and $t$ can be chosen freely. We then use these results to investigate the computational complexity of a problem that arises in computational evolution. Specifically, we show that the problem of deciding if an unrooted binary phylogenetic network has an orientation as a rooted binary phylogenetic network without any shortcuts (the analog of a transitive arcs in phylogenetics) is NP-complete. Our results connect the two (mostly) distinct research areas of orienting undirected graphs and orienting unrooted phylogenetic networks.

发表机构

• University of Auckland（奥克兰大学）

机构由 AI 辅助整理，请以论文原文为准。