Noncrossing Longest Paths and Cycles Noncrossing Longest Paths and Cycles

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...

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Published in:Graphs and combinatorics Vol. 41; no. 6; p. 122
Main Authors: Aloupis, Greg, Biniaz, Ahmad, Bose, Prosenjit, De Carufel, Jean-Lou, Eppstein, David, Maheshwari, Anil, Odak, Saeed, Smid, Michiel, Tóth, Csaba D., Valtr, Pavel
Format: Journal Article
Language:English
Published: Tokyo Springer Japan 01.12.2025
Springer Nature B.V
Subjects:
Combinatorial analysis
Combinatorics
Engineering Design
Graph theory
Graphs
Inequality
Mathematics
Mathematics and Statistics
Motion planning
Original Paper
Questions
ISSN:0911-0119, 1435-5914
Online Access:Get full text
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Summary: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 every 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.
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ISSN:0911-0119
1435-5914
DOI:10.1007/s00373-025-02985-8