Technology Category
Knowledge Representation and Reasoning: Computational Complexity of ReasoningConstraint Satisfaction and Optimization: Other Foundations of Constraint SatisfactionSearch and Optimization: Combinatorial Optimization
Drawing Reeb Graphs
🤖 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
Application Category
📝 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
Research questions and friction points this paper is trying to address.
Exploring complexity of drawing Reeb graphs
Proving NP-hardness of crossing number minimization
Identifying classes with crossing-free drawings
Innovation
Methods, ideas, or system contributions that make the work stand out.
NP-hard proof for Reeb graph crossing minimization
Crossing-free drawings for paths and caterpillars
Optimal algorithm for cycle-shaped Reeb graphs
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