Characterizing Planar Tanglegram Layouts and Applications to Edge Insertion Problems

TL;DR

This paper characterizes all planar layouts of tanglegrams, develops a quadratic-time algorithm for optimal single-edge insertion, and extends results to multiple edge insertions, advancing understanding of tanglegram layout problems.

Contribution

It provides a complete characterization of planar tanglegram layouts and introduces efficient algorithms for edge insertion problems.

Findings

Characterization of all planar tanglegram layouts.

Quadratic-time algorithm for optimal single-edge insertion.

Extension of results to multiple edge insertions.

Abstract

Tanglegrams are formed by taking two rooted binary trees $T$ and $S$ with the same number of leaves and uniquely matching each leaf in $T$ with a leaf in $S$. They are usually represented using layouts, which embed the trees and the matching of the leaves into the plane as in Figure 1. Given the numerous ways to construct a layout, one problem of interest is the Tanglegram Layout Problem, which is to efficiently find a layout that minimizes the number of crossings. This parallels a similar problem involving drawings of graphs, where a common approach is to insert edges into a planar subgraph. In this paper, we will explore inserting edges into a planar tanglegram. Previous results on planar tanglegrams include a Kuratowski Theorem, enumeration, and an algorithm for drawing a planar layout. We start by building on these results and characterizing all planar layouts of a planar tanglegram. We then apply this characterization to construct a quadratic-time algorithm that inserts a single edge optimally. Finally, we generalize some results to multiple edge insertion.