Layer-Respecting Linear Graph Layouts

2026-07-07

AI Summary

This study addresses the layer-respecting linear layout problem, which seeks to minimize edge crossings while preserving a given layer ordering of vertices—critical for effective visualizations such as hierarchical arc diagrams. Recognizing this as an NP-hard problem, the authors propose a fixed-parameter tractable algorithm parameterized by graph width measures (e.g., BFS width). When the BFS width is bounded, the algorithm computes an optimal solution in linear time. By integrating breadth-first search to construct a layered structure with advanced parameterized algorithmic techniques, the method overcomes significant computational bottlenecks and substantially enhances the scalability and efficiency of visualizing large graphs.

Abstract

We show how to visualize a graph, $G=(V,E)$, as a layered drawing, layer-respecting arc diagram, or layer-respecting linear cylindric drawing with a minimum number of edge crossings, where layer-respecting means that layers appear in order on a single line and vertices are grouped by their layers. Even though this problem is NP-hard for general arc diagrams, we show how to create such diagrams with fixed-parameter tractable linear-time algorithms, where the parameter that allows this is the width of a layered graph. Such a layered graph can be obtained from a breadth-first search (BFS), in which case the width is upper bounded by a graph width parameter called the BFS width.

Problem

• layered graph
• edge crossings
• arc diagram
• linear layout
• BFS width

Innovation

• layered graph drawing
• arc diagram
• fixed-parameter tractability
• BFS width
• edge crossing minimization