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

Sabine Cornelsen, Henry Förster, Siddharth Gupta, Stephen Kobourov, Johannes Zink

TL;DR

This work investigates drawing path-based supports of hypergraphs (metro-map visualizations) with an emphasis on bend minimization. It provides a rigorous theoretical framework for straight-line drawings of path-based tree and cactus supports and orthogonal drawings for plane 4-graphs, including NP-hardness for bend-curve minimization, polynomial-time total-bend minimization, and 2-SAT-based zero/one bend decisions. The paper then develops parameterized algorithms: FPT results for curve complexity with respect to the number of through-paths and vertex cover, plus XP results and dynamic-programming approaches extending from trees to cactus supports. It further extends bend-minimization techniques to orthogonal drawings via a Tamassia-style min-cost flow formulation, delivering near-linear-time algorithms under fixed embeddings. The contributions advance both the theoretical understanding and practical algorithmic toolkit for high-quality metro-map visualizations of hypergraphs in restricted graph classes.

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.

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

TL;DR

This work investigates drawing path-based supports of hypergraphs (metro-map visualizations) with an emphasis on bend minimization. It provides a rigorous theoretical framework for straight-line drawings of path-based tree and cactus supports and orthogonal drawings for plane 4-graphs, including NP-hardness for bend-curve minimization, polynomial-time total-bend minimization, and 2-SAT-based zero/one bend decisions. The paper then develops parameterized algorithms: FPT results for curve complexity with respect to the number of through-paths and vertex cover, plus XP results and dynamic-programming approaches extending from trees to cactus supports. It further extends bend-minimization techniques to orthogonal drawings via a Tamassia-style min-cost flow formulation, delivering near-linear-time algorithms under fixed embeddings. The contributions advance both the theoretical understanding and practical algorithmic toolkit for high-quality metro-map visualizations of hypergraphs in restricted graph classes.

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.

Paper Structure

This paper contains 19 sections, 22 theorems, 1 equation, 21 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Straight-Line Drawings of Tree and Cactus Supports
  4. NP-Completeness
  5. Tree Supports -- Exact Algorithms
  6. Tree Supports -- Parameterized Algorithms
  7. Cactus Supports
  8. Orthogonal Drawings for Plane 4-Graph Supports
  9. Open Problems
  10. Acknowledgments.
  11. Disclosure of Interests.
  12. Appendix: Missing Proof Details for NP-Completeness
  13. Appendix: Missing Proof Details for Trees
  14. Appendix: Missing Proofs for Cacti
  15. Linear Cactus Supports
  16. ...and 4 more sections

Key Result

theorem thmcountertheorem

It is NP-complete to decide whether the curve complexity of a path-based tree support is at most $b$ even if

Figures (21)

  • Figure 1: Metro map representations from http://metrosets.ac.tuwien.ac.atmetroMaps:ieee21.
  • Figure 2: Min. the total number of bends and lexicographically max. the bend vector.
  • Figure 3: Lexicographically max. the bend vector.
  • Figure 4: Min. the curve complexity.
  • Figure 5: Min. the curve complexity and the total number of bends.
  • ...and 16 more figures

Theorems & Definitions (43)

  • theorem thmcountertheorem: thm:treeGeneralCurveComplexityHard
  • proof : Overview & Case (i).
  • theorem thmcountertheorem: thm:nphard-max0bendthen1bend
  • theorem thmcountertheorem: thm:treeGeneral-total
  • proof
  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • ...and 33 more