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Flip Distance of Non-Crossing Spanning Trees: NP-Hardness and Improved Bounds

Authors Håvard Bakke Bjerkevik , Joseph Dorfer , Linda Kleist , Torsten Ueckerdt , Birgit Vogtenhuber

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Discrete mathematics
  • Mathematics of computing → Combinatorics
  • Mathematics of computing → Combinatoric problems
Keywords
  • Non-crossing
  • spanning tree
  • plane graph
  • flip graph
  • reconfiguration
  • diameter
  • complexity
  • NP-hard
  • edge exchange
  • compatible flip
  • rotation
  • happy edge property

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Abstract

We consider the problem of reconfiguring non-crossing spanning trees on point sets. For a set P of n points in general position in the plane, the flip graph ℱ(P) has a vertex for each non-crossing spanning tree on P and an edge between any two spanning trees that can be transformed into each other by the exchange of a single edge (coined a flip). This flip graph has been intensively studied, lately with an emphasis on determining its diameter diam(ℱ(P)) for sets P of n points in convex position. For this case, the current best bounds are 14/9⋅n - O(1) ≤ diam(ℱ(P)) < 15/9⋅n - 3, obtained in a recent breakthrough work [Bjerkevik, Kleist, Ueckerdt, and Vogtenhuber; SODA 2025]. The crucial tool for both the upper and lower bound are so-called conflict graphs, which the authors stated might be the key ingredient for determining the diameter (up to lower-order terms).
In this paper, we pick up the concept of conflict graphs from the above-mentioned work and show that this tool is even more versatile than previously hoped. As our first main result, we use conflict graphs to show that computing the flip distance between two non-crossing spanning trees is NP-hard, even for point sets in convex position. Interestingly, the result still holds for more constrained flip operations, concretely, compatible flips (where the removed and the added edge do not cross) and rotations (where the removed and the added edge share an endpoint).
Additionally, we present new insights on the diameter of the flip graph, by this directly extending the line of research from [BKUV SODA25]. Their lower bound is based on a constant-size pair of trees, one of which is of a type we refer to as stacked. We show that if one of the trees is stacked, then the lower bound is indeed optimal up to a constant term, that is, there exists a flip sequence of length at most 14/9⋅(n-1) to any other tree.
Lastly, we improve the lower bound on the diameter of the flip graph ℱ(P) for n points in convex position to 11/7⋅n-o(n).

Cite As Get BibTex

Håvard Bakke Bjerkevik, Joseph Dorfer, Linda Kleist, Torsten Ueckerdt, and Birgit Vogtenhuber. Flip Distance of Non-Crossing Spanning Trees: NP-Hardness and Improved Bounds. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 16:1-16:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.16

Author Details

Håvard Bakke Bjerkevik
  • University at Albany, SUNY, NY, USA
Joseph Dorfer
  • Graz University of Technology, Austria
Linda Kleist
  • University of Hamburg, Germany
Torsten Ueckerdt
  • Karlsruhe Institute of Technology, Germany
Birgit Vogtenhuber
  • Graz University of Technology, Austria

Funding

  • Dorfer, Joseph: Austrian Science Fund (FWF) 10.55776/DOC183.
  • Ueckerdt, Torsten: funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 520708409.
  • Vogtenhuber, Birgit: Austrian Science Fund (FWF) 10.55776/DOC183.

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