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Constrained Flips in Plane Spanning Trees

Authors Oswin Aichholzer , Joseph Dorfer , Birgit Vogtenhuber

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ACM Subject Classification
  • Mathematics of computing → Discrete mathematics
  • Mathematics of computing → Combinatorics
  • Mathematics of computing → Combinatoric problems
Keywords
  • Non-crossing spanning trees
  • Flip Graphs
  • Diameter
  • Complexity
  • Happy edges

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Abstract

A flip in a plane spanning tree T is the operation of removing one edge from T and adding another edge such that the resulting structure is again a plane spanning tree. For trees on a set of points in convex position we study two classic types of constrained flips: (1) Compatible flips are flips in which the removed and inserted edge do not cross each other. We relevantly improve the previous upper bound of 2n-O(√n) on the diameter of the compatible flip graph to (5n/3)-O(1), by this matching the upper bound for unrestricted flips by Bjerkevik, Kleist, Ueckerdt, and Vogtenhuber [SODA 2025] up to an additive constant of 1. We further show that no shortest compatible flip sequence removes an edge that is already in its target position. Using this so-called happy edge property, we derive a fixed-parameter tractable algorithm to compute the shortest compatible flip sequence between two given trees. (2) Rotations are flips in which the removed and inserted edge share a common vertex. Besides showing that the happy edge property does not hold for rotations, we improve the previous upper bound of 2n-O(1) for the diameter of the rotation graph to (7n/4)-O(1).

Cite As Get BibTex

Oswin Aichholzer, Joseph Dorfer, and Birgit Vogtenhuber. Constrained Flips in Plane Spanning Trees. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 5:1-5:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.GD.2025.5

Author Details

Oswin Aichholzer
  • Graz University of Technology, Austria
Joseph Dorfer
  • Graz University of Technology, Austria
Birgit Vogtenhuber
  • Graz University of Technology, Austria

Funding

  • Dorfer, Joseph: Austrian Science Foundation (FWF) 10.55776/DOC183.

References

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