Two-Layer Drawings with a Tree on Top: Vertex Splits and Fixed-Parameter Algorithms

2026-07-27

AI Summary

This work addresses the problem of crossing-free drawing of bipartite graphs under auxiliary tree constraints, where the ordering of vertices on one layer must conform to a given tree structure, and the goal is to eliminate all edge crossings by minimizing vertex splits. We propose two single-exponential fixed-parameter tractable algorithms parameterized respectively by the number of splits and the maximum degree of the constraint tree, and show that their running times are tight under the Exponential Time Hypothesis (ETH). Notably, the algorithm parameterized by the tree’s maximum degree extends to the classical unconstrained setting, yielding a significant improvement over existing results. Both theoretical analysis and experimental evaluation demonstrate the superiority of our approaches in terms of time complexity and practical performance.

Abstract

Two-layer drawings of bipartite graphs place the vertices of each part on one of two parallel lines and draw the edges as straight-line links. Traditionally, the optimization goal is to find vertex permutations on one or both layers that minimize the induced number of edge crossings. This problem is NP-hard, and crossing-minimal solutions may still contain many crossings. Recently, there has been growing interest in an orthogonal optimization goal, namely removing all crossings by vertex splitting, i.e., replacing original vertices by two or more copies and distributing the adjacencies among them. In this paper, we study a natural extension of the two-layer vertex splitting problem in which the vertex order on one layer is constrained by a given auxiliary tree $T$, motivated by applications such as the visualization of anatomical hierarchies in the Human Reference Atlas. We investigate the parameterized complexity of this problem and obtain two main contributions: (1) a fixed-parameter algorithm with respect to the number $k$ of splits, and (2) an ETH-tight single-exponential fixed-parameter algorithm with respect to the maximum degree of $T$. Moreover, we build on the latter result to obtain an ETH-tight single-exponential algorithm for the classical unconstrained version of the problem, improving upon the previous $O^*(2^{k\cdot \log k})$ algorithms. Finally, we also implement our algorithm and show that it performs well in practice.

Problem

• two-layer drawings
• vertex splitting
• crossing removal
• parameterized complexity
• tree constraints

Innovation

• vertex splitting
• fixed-parameter algorithm
• two-layer drawing
• parameterized complexity
• ETH-tight