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Tanglegrams are formed by taking two rooted binary trees $T$ and $S$ with thesame 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 thematching of the leaves into the plane as in Figure 1. Given the numerous waysto construct a layout, one problem of interest is the Tanglegram LayoutProblem, which is to efficiently find a layout that minimizes the number ofcrossings. This parallels a similar problem involving drawings of graphs, wherea common approach is to insert edges into a planar subgraph. In this paper, wewill explore inserting edges into a planar tanglegram. Previous results onplanar tanglegrams include a Kuratowski Theorem, enumeration, and an algorithmfor drawing a planar layout. We start by building on these results andcharacterizing all planar layouts of a planar tanglegram. We then apply thischaracterization to construct a quadratic-time algorithm that inserts a singleedge optimally. Finally, we generalize some results to multiple edge insertion.

Characterizing Planar Tanglegram Layouts and Applications to Edge Insertion Problems