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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]
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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]
ISSN:01795376
DOI:10.1007/s00454-025-00785-8