Noncrossing Longest Paths and Cycles

TL;DR

The paper addresses whether the longest spanning path or cycle on a finite planar point set must include a crossing edge and provides a negative answer by constructing point sets for which the longest path and the longest cycle are noncrossing and unique. The authors introduce a perturbation framework that starts from a one‑dimensional base set $P$ on the $x$-axis and lifts it to a plane set $P'$ with small vertical offsets, so that the longest structure on $P'$ corresponds to a longest structure on $P$ while becoming unique and noncrossing. They develop explicit constructions for both even and odd numbers of points for paths and cycles, and extend the framework to longest matchings, all while establishing useful structural properties of such longest objects. The results settle open questions about the crossing behavior of maximal-length geometric graphs and provide a versatile method for generating large noncrossing configurations with provable maximality properties, with potential implications for graph drawing and combinatorial geometry.

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, Álvarez-Rebollar, Cravioto-Lagos, Marín, Solé-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.

Figures (8)

• Figure 1: Illustration of a longest path for a point set on a line, for the case where the number of points, $n$, is even. Numbers below intervals $[p_{n/2+i},p_{n/2+i+1}]$ represent the multiplicity of the contribution of the corresponding intervals to the length of the longest path.
• Figure 2: Illustration of the construction of a longest path for $2k$ points. The figure is not true to scale as the real $y$-coordinates are very small so that the points lie almost on the $x$-axis. (a) Lifting $(x_1,0)$ to the $y$-coordinate $\frac{\beta}{8k}$. (b) The final longest path.
• Figure 3: Illustration of the construction of a longest cycle for $4k{-}2$ points. The figure is not true to scale. The $y$-coordinates should be small enough so that all points lie almost on the $x$-axis.
• Figure 4: The longest cycle connects $p_{-1}$ to $p_{2}$ and $p'_{-1}$ to $p'_{2}$
• Figure 5: Illustration of the construction of a longest cycle for $2k{+}1$ points.

Theorems & Definitions (34)

• Conjecture 1: Rebollar2024
• Lemma 1
• proof
• Corollary 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4