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Reconfiguration of Plane Trees in Convex Geometric Graphs.

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Title: Reconfiguration of Plane Trees in Convex Geometric Graphs.
Authors: Bousquet, Nicolas1 (AUTHOR) nicolas.bousquet@cnrs.fr, De Meyer, Lucas1 (AUTHOR) lucas.de-meyer@univ-lyon1.fr, Pierron, Théo1 (AUTHOR) theo.pierron@univ-lyon1.fr, Wesolek, Alexandra1 (AUTHOR) alexandra_wesolek@sfu.ca
Source: Discrete & Computational Geometry. Mar2026, Vol. 75 Issue 2, p431-464. 34p.
Subjects: Spanning trees, Edges (Geometry), Geometric vertices
Abstract: A non-crossing spanning tree of a set of points in the plane is a spanning tree whose edges pairwise do not cross. Avis and Fukuda in 1996 proved that there always exists a flip sequence of length at most 2 n - 4 between any pair of non-crossing spanning trees (where n denotes the number of points). Hernando et al. proved that the length of a minimal flip sequence can be of length at least 3 2 n . Two recent results of Aichholzer et al. and Bousquet et al. improved the upper bound by Avis and Fukuda by proving that there always exists a flip sequence of length respectively at most 2 n - log n and 2 n - n when the points are in convex position. We pursue the investigation of the convex case by improving the upper bound by a linear factor for the first time in 30 years. We prove that there always exists a flip sequence between any pair of non-crossing spanning trees T 1 , T 2 of length at most cn where c ≈ 1.95 . Our result is actually stronger since we prove that, for any two trees T 1 , T 2 , there exists a flip sequence from T 1 to T 2 of length at most c | T 1 \ T 2 | . We also improve the best lower bound in terms of the symmetric difference by proving that there exists a pair of trees T 1 , T 2 such that a minimal flip sequence has length 5 3 | T 1 \ T 2 | , improving the lower bound of Hernando et al. by considering the symmetric difference instead of the number of vertices. We generalize this lower bound construction to non-crossing flips (where we close the gap between upper and lower bounds) and edge-rotations. [ABSTRACT FROM AUTHOR]
Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: <searchLink fieldCode="JN" term="%22Discrete+%26+Computational+Geometry%22">Discrete & Computational Geometry</searchLink>. Mar2026, Vol. 75 Issue 2, p431-464. 34p.
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  Data: A non-crossing spanning tree of a set of points in the plane is a spanning tree whose edges pairwise do not cross. Avis and Fukuda in 1996 proved that there always exists a flip sequence of length at most 2 n - 4 between any pair of non-crossing spanning trees (where n denotes the number of points). Hernando et al. proved that the length of a minimal flip sequence can be of length at least 3 2 n . Two recent results of Aichholzer et al. and Bousquet et al. improved the upper bound by Avis and Fukuda by proving that there always exists a flip sequence of length respectively at most 2 n - log n and 2 n - n when the points are in convex position. We pursue the investigation of the convex case by improving the upper bound by a linear factor for the first time in 30 years. We prove that there always exists a flip sequence between any pair of non-crossing spanning trees T 1 , T 2 of length at most cn where c ≈ 1.95 . Our result is actually stronger since we prove that, for any two trees T 1 , T 2 , there exists a flip sequence from T 1 to T 2 of length at most c | T 1 \ T 2 | . We also improve the best lower bound in terms of the symmetric difference by proving that there exists a pair of trees T 1 , T 2 such that a minimal flip sequence has length 5 3 | T 1 \ T 2 | , improving the lower bound of Hernando et al. by considering the symmetric difference instead of the number of vertices. We generalize this lower bound construction to non-crossing flips (where we close the gap between upper and lower bounds) and edge-rotations. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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              Text: Mar2026
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