# Simultaneous Drawing of Layered Trees

## TL;DR

This paper investigates the crossing minimization problem in layered graph drawings of rooted trees, providing polynomial-time solutions for two trees and an XP-time algorithm for multiple trees with three layers.

## Contribution

It introduces a dynamic programming approach for two trees and reduces the problem to shortest path for multiple trees with three layers.

## Key findings

- Polynomial-time algorithm for two trees.
- XP-time algorithm for multiple trees with three layers.
- Reduction to shortest path problem.

## Abstract

We study the crossing-minimization problem in a layered graph drawing of planar-embedded rooted trees whose leaves have a given total order on the first layer, which adheres to the embedding of each individual tree. The task is then to permute the vertices on the other layers (respecting the given tree embeddings) in order to minimize the number of crossings. While this problem is known to be NP-hard for multiple trees even on just two layers, we describe a dynamic program running in polynomial time for the restricted case of two trees. If there are more than two trees, we restrict the number of layers to three, which allows for a reduction to a shortest-path problem. This way, we achieve XP-time in the number of trees.