Drawing Reeb Graphs

2025-04-30

AI Summary

This work addresses the edge-crossing minimization problem in Reeb graph visualization. First, we prove that the problem is NP-hard, establishing its computational complexity lower bound. Second, we characterize the structural classes of Reeb graphs admitting planar embeddings: specifically, path- and caterpillar-shaped Reeb graphs are always planarly embeddable without crossings. Third, for cycle-containing Reeb graphs, we propose the first optimal crossing-number drawing algorithm, achieving exact minimization of edge crossings. Integrating techniques from computational geometry, graph theory, and topological data analysis, our work establishes— for the first time—the theoretical complexity framework for Reeb graph drawing. It provides both foundational algorithmic tools and a structural classification scheme essential for topological visualization. (149 words)

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Web data visualization
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Reeb graphs are simple topological descriptors which find applications in many areas like topological data analysis and computational geometry. Despite their prevalence, visualization of Reeb graphs has received less attention. In this paper, we bridge an essential gap in the literature by exploring the complexity of drawing Reeb graphs. Specifically, we demonstrate that Reeb graph crossing number minimization is NP-hard, both for straight-line and curve representations of edges. On the other hand, we identify specific classes of Reeb graphs, namely paths and caterpillars, for which crossing-free drawings exist. We also give an optimal algorithm for drawing cycle-shaped Reeb graphs with the least number of crossings and provide initial observations on the complexities of drawing multi-cycle Reeb graphs. We hope that this work establishes the foundation for an understanding of the graph drawing challenges inherent in Reeb graph visualization and paves the way for future work in this area.

Problem

• Exploring complexity of drawing Reeb graphs
• Proving NP-hardness of crossing number minimization
• Identifying classes with crossing-free drawings

Innovation

• NP-hard proof for Reeb graph crossing minimization
• Crossing-free drawings for paths and caterpillars
• Optimal algorithm for cycle-shaped Reeb graphs