Drawing Tree-Based Phylogenetic Networks with Minimum Number of Crossings

TL;DR

This paper investigates the problem of minimizing crossings in visualizing tree-based phylogenetic networks, proving NP-completeness for x-monotone reticulation edges and providing efficient solutions for ear-shaped edges.

Contribution

It introduces the first formal study of crossing minimization in phylogenetic network visualization, establishing complexity results and algorithms for different edge drawing styles.

Findings

NP-complete for x-monotone reticulation edges

Fixed-parameter tractable in the number of reticulations

Quadratic-time solution for ear-shaped reticulation edges

Abstract

In phylogenetics, tree-based networks are used to model and visualize the evolutionary history of species where reticulate events such as horizontal gene transfer have occurred. Formally, a tree-based network $N$ consists of a phylogenetic tree $T$ (a rooted, binary, leaf-labeled tree) and so-called reticulation edges that span between edges of $T$. The network $N$ is typically visualized by drawing $T$ downward and planar and reticulation edges with one of several different styles. One aesthetic criteria is to minimize the number of crossings between tree edges and reticulation edges. This optimization problem has not yet been researched. We show that, if reticulation edges are drawn x-monotone, the problem is NP-complete, but fixed-parameter tractable in the number of reticulation edges. If, on the other hand, reticulation edges are drawn like "ears", the crossing minimization problem can be solved in quadratic time.