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
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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 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.
AbstractList	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. 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.
ArticleNumber	122
Author	Biniaz, Ahmad Valtr, Pavel Smid, Michiel Bose, Prosenjit De Carufel, Jean-Lou Odak, Saeed Aloupis, Greg Maheshwari, Anil Tóth, Csaba D. Eppstein, David
Author_xml	– sequence: 1 givenname: Greg surname: Aloupis fullname: Aloupis, Greg organization: Khoury College of Computer Sciences, Northeastern University – sequence: 2 givenname: Ahmad surname: Biniaz fullname: Biniaz, Ahmad email: abiniaz@uwindsor.ca organization: School of Computer Science, University of Windsor – sequence: 3 givenname: Prosenjit surname: Bose fullname: Bose, Prosenjit organization: School of Computer Science, Carleton University – sequence: 4 givenname: Jean-Lou surname: De Carufel fullname: De Carufel, Jean-Lou organization: School of Electrical Engineering and Computer Science, University of Ottawa – sequence: 5 givenname: David surname: Eppstein fullname: Eppstein, David organization: Computer Science Department, University of California – sequence: 6 givenname: Anil surname: Maheshwari fullname: Maheshwari, Anil organization: School of Computer Science, Carleton University – sequence: 7 givenname: Saeed surname: Odak fullname: Odak, Saeed organization: School of Electrical Engineering and Computer Science, University of Ottawa – sequence: 8 givenname: Michiel surname: Smid fullname: Smid, Michiel organization: School of Computer Science, Carleton University – sequence: 9 givenname: Csaba D. surname: Tóth fullname: Tóth, Csaba D. organization: Department of Mathematics, California State University Northridge, Department of Computer Science, Tufts University – sequence: 10 givenname: Pavel surname: Valtr fullname: Valtr, Pavel organization: Department of Applied Mathematics, Charles University
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SubjectTerms	Combinatorial analysis Combinatorics Engineering Design Graph theory Graphs Inequality Mathematics Mathematics and Statistics Motion planning Original Paper Questions
Subtitle	Noncrossing Longest Paths and Cycles
Title	Noncrossing Longest Paths and Cycles
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