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Structural Properties of Shortest Flip Sequences Between Plane Spanning Trees

Authors Oswin Aichholzer , Joseph Dorfer , Peter Kramer , Christian Rieck , Birgit Vogtenhuber

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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Spanning trees
  • convex position
  • non-crossing
  • flip graph
  • happy edges
  • parking conjecture

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Abstract

We study the problem of reconfiguring plane spanning trees on point sets in the plane in convex position. A flip is a reconfiguration step in which one edge of a tree is replaced with another, yielding again a plane spanning tree. For a given point set P, the flip graph has a vertex for each plane spanning tree on P, and an edge between any two trees that differ by a single flip. The flip distance between two trees is the length of a shortest path between them in the flip graph, that is, the minimum number of flips needed to transform one tree into the other.
We study structural properties of shortest flip sequences. Over the years, several conjectures have been proposed regarding these sequences. The folklore happy edge conjecture suggests that any edge shared by both the initial and target tree is never flipped in a shortest flip sequence. Moreover, the parking conjecture claims that in shortest flip sequences, every edge is either flipped to a target edge or parked on the convex hull. Notably, the happy edge conjecture is implied by the parking conjecture. Furthermore, it has been conjectured that an edge is never reparked, that is, any edge resulting from a flip either is a target edge or is subsequently flipped to a target edge. Essentially, all recent flip algorithms rely on these three conjectures and the properties they imply.
In this paper, we disprove both the parking and the reparking conjecture. In particular, we show that a flip sequence obeying the parking property sometimes needs linearly more flips than a shortest flip sequence, and exhibit examples where any shortest flip sequence flips some edge multiple times. In contrast, we identify several conditions under which the parking and reparking properties hold; these imply that the reparking property holds for compatible flips, and may be exploited by algorithms to obtain short flip sequences, especially for the compatible case.

Cite As Get BibTex

Oswin Aichholzer, Joseph Dorfer, Peter Kramer, Christian Rieck, and Birgit Vogtenhuber. Structural Properties of Shortest Flip Sequences Between Plane Spanning Trees. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 37:1-37:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.GD.2026.37

Author Details

Oswin Aichholzer
  • Institute of Algorithms and Theory, Graz University of Technology, Austria
Joseph Dorfer
  • Institute of Algorithms and Theory, Graz University of Technology, Austria
Peter Kramer
  • Department of Computer Science, TU Braunschweig, Germany
Christian Rieck
  • Institute of Mathematics, University of Kassel, Germany
Birgit Vogtenhuber
  • Institute of Algorithms and Theory, Graz University of Technology, Austria

Funding

  • Dorfer, Joseph: Austrian Science Fund (FWF) 10.55776/DOC183.
  • Kramer, Peter: Partially funded by a fellowship of the German Academic Exchange Service (DAAD) and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), grant 530918134.
  • Rieck, Christian: Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), grant 522790373.

Acknowledgements

This work was initiated at the Reconfiguration Week hosted at TU Graz in March 2025. We thank the organizers and participants for fruitful discussions and a good atmosphere.

Supplementary Materials

References

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