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# Reconfiguration of Plane Trees in Convex Geometric Graphs

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LIPIcs.SoCG.2024.22.pdf
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## Acknowledgements

The first and third authors would like to thank Valentin Gledel for interesting discussions on the problems on an earlier stage of this project.

## Cite As

Nicolas Bousquet, Lucas de Meyer, Théo Pierron, and Alexandra Wesolek. Reconfiguration of Plane Trees in Convex Geometric Graphs. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 22:1-22:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SoCG.2024.22

## 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 2n-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 Avis and Fukuda upper bound by proving that there always exists a flip sequence of length respectively at most 2n-log n and 2n-√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 T₁,T₂ of length at most c n where c ≈ 1.95. Our result is actually stronger since we prove that, for any two trees T₁,T₂, there exists a flip sequence from T₁ to T₂ of length at most c |T₁ ⧵ T₂|. We also improve the best lower bound in terms of the symmetric difference by proving that there exists a pair of trees T₁,T₂ such that a minimal flip sequence has length (5/3) |T₁ ⧵ T₂|, 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 rotations.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Trees
##### Keywords
• Reconfiguration
• Non-crossing trees
• flip distance

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## References

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