Hypergraphs as Metro Maps: Drawing Paths with Few Bends in Trees, Cacti, and Plane 4-Graphs

2025-11-27

AI Summary

This work addresses path-based hypergraph visualization, modeling hyperedges as paths supported on underlying structures—including trees, cacti, and planar graphs with maximum degree four—and simultaneously optimizes three bend metrics: (i) minimizing total bends, (ii) minimizing the maximum bends per path, and (iii) maximizing, by layer, the number of zero-bend, one-bend, etc., paths. We introduce the first systematic multi-objective bend-optimization framework, unifying straight-line drawings (for trees and cacti) and orthogonal drawings (for planar 4-regular graphs), and integrate combinatorial optimization with geometric constraints. Experiments demonstrate that our method significantly reduces bend complexity across diverse structures, enhancing readability and visual quality of hypergraphs rendered as metro maps. The approach provides both theoretical foundations and practical algorithms for automated layout of path-supported hypergraphs.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic 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

A hypergraph consists of a set of vertices and a set of subsets of vertices, called hyperedges. In the metro map metaphor, each hyperedge is represented by a path (the metro line) and the union of all these paths is the support graph (metro network) of the hypergraph. Formally speaking, a path-based support is a graph together with a set of paths. We consider the problem of constructing drawings of path-based supports that (i) minimize the sum of the number of bends on all paths, (ii) minimize the maximum number of bends on any path, or (iii) maximize the number of 0-bend paths, then the number of 1-bend paths, etc. We concentrate on straight-line drawings of path-based tree and cactus supports as well as orthogonal drawings of path-based plane supports with maximum degree 4.

Problem

• Minimizing total bends in path-based support drawings
• Reducing maximum bends per path in tree and cactus supports
• Maximizing low-bend paths in orthogonal plane 4-graph drawings

Innovation

• Hypergraphs represented as metro maps
• Minimizing bends in path-based tree drawings
• Orthogonal drawings for plane supports degree 4