Noncrossing Longest Paths and Cycles

TL;DR

This paper demonstrates that the longest geometric structures such as paths and cycles on a set of points can be noncrossing, countering previous assumptions and conjectures about their crossing properties.

Contribution

We provide a framework to construct arbitrarily large point sets where the longest paths and cycles are noncrossing, refuting prior conjectures about their crossing behavior.

Findings

Longest spanning paths can be noncrossing.

Longest spanning cycles can be noncrossing.

Counterexamples to previous conjectures.

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.