A characterization of terminal planar networks by forbidden structures

TL;DR

This paper characterizes terminal planar networks through forbidden structures, providing finite criteria and efficient algorithms for testing planarity, with applications beyond phylogenetics and connections to upward planarity problems.

Contribution

It introduces a Kuratowski-type theorem for terminal planar networks using forbidden structures and offers linear-time algorithms for planarity testing and drawing.

Findings

Finite set of forbidden structures characterizes terminal planar networks

Linear-time planarity testing algorithms developed

Applications extend beyond phylogenetics

Abstract

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.