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Computer Science > Computational Geometry

arXiv:2602.22874 (cs)
[Submitted on 26 Feb 2026]

Title:Flip Distance of Triangulations of Convex Polygons / Rotation
Distance of Binary Trees is NP-complete

Authors:Joseph Dorfer
View a PDF of the paper titled Flip Distance of Triangulations of
Convex Polygons / Rotation Distance of Binary Trees is NP-complete,
by Joseph Dorfer
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    Abstract:Flips in triangulations of convex polygons arise in many
    different settings. They are isomorphic to rotations in binary
    trees, define edges in the 1-skeleton of the Associahedron and
    cover relations in the Tamari Lattice.
    The complexity of determining the minimum number of flips that
    transform one triangulation of a convex point set into another
    remained a tantalizing open question for many decades. We settle
    this question by proving that computing shortest flip sequences
    between triangulations of convex polygons, and therefore also
    computing the rotation distance of binary trees, is NP-hard.
    For our proof we develop techniques for flip sequences of
    triangulations whose counterparts were introduced for the study
    of flip sequences of non-crossing spanning trees by Bjerkevik,
    Kleist, Ueckerdt, and Vogtenhuber~[SODA25] and Bjerkevik, Dorfer,
    Kleist, Ueckerdt, and Vogtenhuber~[SoCG26].

          Computational Geometry (cs.CG); Computational Complexity
Subjects: (cs.CC); Discrete Mathematics (cs.DM); Data Structures and
          Algorithms (cs.DS); Combinatorics (math.CO)
Cite as:  arXiv:2602.22874 [cs.CG]
          (or arXiv:2602.22874v1 [cs.CG] for this version)
          https://doi.org/10.48550/arXiv.2602.22874
          Focus to learn more
          arXiv-issued DOI via DataCite (pending registration)

Submission history

From: Joseph Dorfer [view email]
[v1] Thu, 26 Feb 2026 11:10:47 UTC (435 KB)
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    NP-complete, by Joseph Dorfer
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