Haruki Miyaji | Searcharxiv

arXiv2026

A characterization of terminal planar networks by forbidden structures

The class of terminal planar networks was recently introduced from a biological perspective in relation to the visualization of phylogenetic networks, and its connection to upward planar networks has been established. We provide a Kuratowski-type theorem that characterizes terminal planar networks by a finite set of forbidden structures, defined via six families of 0/1-labeled graphs. Another characterization based on planarity of supergraphs yields linear-time algorithms for testing terminal planarity and for computing such planar drawings. We describe an application that is potentially relevant in broader, non-phylogenetic settings. We also discuss a connection of our main result to an open problem on the forbidden structures of single-source upward planar networks.

math.CO↗

arXiv2025

Orientability of Undirected Phylogenetic Networks to a Desired Class: Practical Algorithms and Application to Tree-Child Orientation

The C-Orientation problem asks whether it is possible to orient an undirected graph to a directed phylogenetic network of a desired network class C. This problem arises, for example, when visualising evolutionary data, as popular methods such as Neighbor-Net are distance-based and inevitably produce undirected graphs. The complexity of C-Orientation remains open for many classes C, including binary tree-child networks, and practical methods are still lacking. In this paper, we propose an exact FPT algorithm for C-Orientation that is applicable to any class C and parameterised by the reticulation number and the maximum size of minimal basic cycles, and a very fast heuristic for Tree-Child Orientation. While the state-of-the-art for C-Orientation is a simple exponential time algorithm whose computational bottleneck lies in searching for appropriate reticulation vertex placements, our methods significantly reduce this search space. Experiments show that, although our FPT algorithm is still exponential, it significantly outperforms the existing method. The heuristic runs even faster but with increasing false negatives as the reticulation number grows. Given this trade-off, we also discuss theoretical directions for improvement and biological applicability of the heuristic approach.

cs.DS↗