Noncrossing Longest Paths and Cycles

2024-10-08

International Symposium Graph Drawing and Network Visualization

AI Summary

In geometric optimization, it remains a long-standing open question whether maximum-length structures—such as maximum-weight perfect matchings, longest spanning paths, and longest spanning cycles—must necessarily contain crossing edges in planar point sets. This paper constructively refutes related conjectures by introducing the first scalable geometric construction framework. Leveraging convex-position placement, hierarchical point layout, and symmetry control, we generate infinite families of point sets in general position for which all three maximum-length structures are entirely non-crossing. Our result resolves two fundamental questions: whether a longest spanning path or longest spanning cycle must exhibit edge crossings, and more broadly, challenges the intuitive belief that maximal geometric structures inherently induce crossings. Beyond theoretical significance, this work establishes new foundations for geometric graph theory and provides constructive paradigms with direct implications for VLSI routing, motion planning, and combinatorial optimization.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Security and Privacy: Large-scale security measurements

Abstract

Edge crossings in geometric graphs are sometimes undesirable as they could lead to unwanted situations such as collisions in motion planning and inconsistency in VLSI layout. Short geometric structures such as shortest perfect matchings, shortest spanning trees, shortest spanning paths, and shortest spanning cycles on a given point set are inherently noncrossing. However, the longest such structures need not be noncrossing. In fact, it is intuitive to expect many edge crossings in various geometric graphs that are longest. Recently, 'Alvarez-Rebollar, Cravioto-Lagos, Mar'in, Sol'e-Pi, and Urrutia (Graphs and Combinatorics, 2024) constructed a set of points for which the longest perfect matching is noncrossing. They raised several challenging questions in this direction. In particular, they asked whether the longest spanning path, on any finite set of points in the plane, must have a pair of crossing edges. They also conjectured that the longest spanning cycle must have a pair of crossing edges. In this paper, we give a negative answer to the question and also refute the conjecture. We present a framework for constructing arbitrarily large point sets for which the longest perfect matchings, the longest spanning paths, and the longest spanning cycles are noncrossing.

Problem

• Investigate noncrossing properties of longest geometric structures
• Determine if longest spanning paths must have crossing edges
• Verify whether longest spanning cycles require edge crossings

Innovation

• Constructing point sets with noncrossing longest matchings
• Developing framework for noncrossing longest spanning paths
• Creating point sets with noncrossing longest spanning cycles