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

Published 26 Feb 2026 in cs.CG, cs.CC, cs.DM, cs.DS, and math.CO | (2602.22874v1)

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].

Summary

  • The paper proves that determining the flip distance between convex polygon triangulations and the corresponding rotation distance in binary trees is NP-complete.
  • The authors employ a reduction using conflict graphs, gadget constructions, and a blow-up operation to encode a planar monotone Max-2SAT instance.
  • This breakthrough resolves a long-standing open problem, impacting computational geometry, data structures, and polyhedral combinatorics.

NP-Completeness of Flip Distance and Rotation Distance

Catalan Structures and Reconfiguration Operations

The paper “Flip Distance of Triangulations of Convex Polygons / Rotation Distance of Binary Trees is NP-complete” (2602.22874) addresses a long-standing open problem concerning the computational complexity of determining minimal flip (rotation) sequences in Catalan-counted combinatorial structures. These include triangulations of convex polygons, rotations in binary trees, and various bracketings. Such structures admit elementary reconfiguration steps—diagonal flips in triangulations, rotations in trees, or bracket rearrangement—whose induced graphs have rich combinatorial geometry and algebraic structure, notably the Tamari lattice and the Associahedron's 1-skeleton.

The paper focuses specifically on the problem of computing the shortest flip sequence that transforms one triangulation of a convex polygon into another, equivalently the rotation distance between two binary trees. The bijection between these objects allows the central result to hold for both domains.

Main Results: NP-Completeness

The central claim is that deciding whether the flip distance between two triangulations of a convex polygon (or, the rotation distance between two binary trees) is at most kk is NP-complete. This is contradictory to prior conjectures and expectations, which had been shaped by the happy edge property in convex cases and the existence of strong upper bounds on flip graph diameter (e.g., $2n-10$ flips for nn-gons) [sleator1986rotation, pournin2014diameter]. Previously, NP-hardness results existed for point sets in general position, polygons with holes, and simple polygons, but convex polygons were widely believed to possess exploitable structure yielding tractability [lubiw2015flip, pilz2014flip, aichholzer2015flip]. The paper proves otherwise, closing this gap.

Methodology and Technical Framework

The reduction uses nuanced gadgets inspired by recent work on non-crossing spanning tree flips [Trees2026hard]. The technical innovations include:

  • Linear Representations: Convex triangulations are reformulated on a linear spine, facilitating gadget placements and rigorous control of crossing relations.
  • Blow-up Operation: Edges are subdivided and "fans" of triangles are constructed, introducing a parameter β\beta that controls the hardness amplification.
  • Conflict Graphs: Directed graphs are defined to encode the dependencies and obstructions between triangle pairings in flip sequences. The reduction focuses on finding maximum acyclic subsets in these conflict graphs.
  • Encoding Planar Separable Monotone Max-2SAT: The conflict graph acyclicity problem is reduced from a planar monotone Max-2SAT instance, mirrored in the geometric configuration of variable, clause, and interaction gadgets along the spine.

The proof compiles carefully synchronized upper and lower bounds on the flip distance, formulated in terms of the conflict graph's acyclic subset cardinality. For sufficiently large β\beta, the dominating terms are linear in β\beta, capturing the NP-hardness signature.

Binary Trees and Rotation Distance

Given the established isomorphism between the flip graph of triangulations and the rotation graph of binary trees [sleator1986rotation], the result holds for rotation distance: computing the minimal number of rotations between binary trees is also NP-complete. This resolves the classical question posed by Culik and Wood [CULIK198239].

Generalizations to Lattices and Polytopes

The result propagates to diverse Catalan-derived structures:

  • Tamari Lattice and Bracketings: Shortest path determination along cover relations in the Tamari and its generalizations (framing lattice, Cambrian lattices, permutree lattice, etc.) is NP-hard.
  • Associahedra and Polytope Generalizations: The NP-hardness extends to the 1-skeleton of generalized associahedra, brick polytopes, permutohedra, quotientopes, and multiassociahedra [ceballos2015many, MR1941227, MR2820759, MR3964495, MR3221537]. Even some polyhedral extensions containing flip graphs as subgraphs inherit hardness under mild structural conditions.

Plane Graph Reconfiguration

For pseudo-triangulations, the flip distance between pointed pseudo-triangulations is NP-complete, reflecting the same hardness for the convex case. Notably, perfect matchings and paths on convex point sets remain tractable, indicating sharp demarcations in the combinatorial landscape.

Algorithmic Boundaries and Parameterized Complexity

While the general problem is NP-complete, there exist significant advances in algorithmic bounds:

  • FPT Algorithms: Parameterized approaches exploit topological orderings and happy edge properties yielding exponential-time algorithms for small distances [li20253, kanj2017computing, feng2021improved, lucas2010improved]. The best-known FPT algorithm for convex flip distance operates in O(3.82k)O(3.82^k) time [li20253], where kk is the flip distance.
  • Exact and Approximate Bounds: Polynomial upper and lower bounds, as well as specific combinatorial bounds based on crossings, have been derived for restricted cases [baril2006efficient, hanke1996edge, eppstein2010happy].

The hardness result categorically eliminates the prospect of a polynomial-time algorithm for the general convex polygon flip distance problem unless P=NP\text{P} = \text{NP}.

Theoretical and Practical Implications

The formal intractability profoundly impacts several branches:

  • Combinatorial Geometry: The result provides closure on the complexity of fundamental reconfiguration problems in geometric combinatorics.
  • Data Structures: In computer science, rotation distance in binary trees connects directly to balancing algorithms (AVL, splay trees, etc.), heightening the relevance for tree edit distance applications.
  • Algebraic and Polyhedral Combinatorics: The propagation of NP-hardness to cover relations in Tamari-type lattices and skeletons of polyhedral structures guides expectations for algorithmic feasibility in generalized cluster algebras and related frameworks.

The explicit reduction techniques and conflict graph formulations are likely to stimulate further exploration of combinatorial reconfiguration problems—especially for generalizations where tractability remains uncharacterized.

Future Directions

The dichotomy between convex and non-convex configurations, or Catalan-structured vs. non-Catalan, suggests fertile ground for nuanced complexity classifications. There remain open problems in convex perfect matchings and path flips, as well as shortest path questions in larger or hybrid polyhedral structures (e.g., cosmohedra, permuto-associahedra). The interplay of combinatorial, geometric, and algebraic perspectives promises continued developments in both algorithmic and structural theory.

Conclusion

By proving the NP-completeness of computing flip distance between triangulations of convex polygons and rotation distance of binary trees, the paper firmly establishes the inherent computational intractability of these canonical Catalan reconfiguration problems (2602.22874). The result closes a decades-old problem and decisively informs both theoretical understanding and practical algorithm design in combinatorial geometry, data structures, and algebraic combinatorics. The techniques developed—particularly the linear representation and conflict graph constructions—provide robust tools for further study across related domains.

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What this paper is about (in simple terms)

This paper studies a puzzle-like problem: you have a shape (a convex polygon) split into triangles, and you want to change it into another triangle-splitting using the fewest possible tiny moves. Each tiny move, called a “flip,” replaces one diagonal inside a small four-sided area with the other diagonal. The big discovery of the paper is that figuring out the minimum number of flips needed is an NP-complete problem. That means it is very likely a “hard” problem that no one can solve quickly in general.

This result also applies to binary trees (a common computer science structure). There, the tiny move is called a “rotation,” and the minimum number of rotations to change one tree into another is also NP-complete to compute.

The main questions the paper answers

  • How hard is it to compute the minimum number of flips needed to turn one triangulation of a convex polygon into another?
  • Equivalently, how hard is it to compute the minimum number of rotations to turn one binary tree into another?

The paper proves the answer: deciding if there is a sequence of at most k flips (or rotations) is NP-complete.

How the authors approach the problem

To explain the approach, here are the key ideas in everyday language:

  • Triangulations and flips: Picture points around a circle connected by straight lines that don’t cross, splitting the inside into triangles. A flip is like swapping the diagonal of a small quadrilateral (a “diamond” shape).
  • Binary trees and rotations: Binary trees are branching diagrams where each internal node splits into two branches. A rotation is a small, local reattachment of branches. There is a perfect one-to-one correspondence between flips in triangulations and rotations in binary trees.
  • A helpful picture (linear representation): Instead of drawing points on a circle, the authors lay all points on a straight line (a “spine”) and draw arcs above and below it. This makes it easier to see and reason about which parts “block” each other.
  • Conflict graph (who blocks whom): The authors build a directed graph that records “traffic rules” between groups of edges. An arrow A → B means: you can’t put in the B-edges until you’ve removed the A-edges, because they’d cross. This graph captures the essential bottlenecks in any flip sequence.
  • Blow-ups (turning lanes into multi-lane highways): To make “blocking” count clearly in the total number of flips, they replace certain single edges with bundles of many parallel edges (a “fan”). Think of it like widening a road so that any delay becomes very obvious in the total travel time. By choosing the bundle size big enough, the total number of flips mostly depends on the structure of the conflict graph.
  • Gadgets and a reduction from a known hard problem: To prove NP-hardness, they start from a well-known hard logic puzzle (a special planar version of Max-2SAT) and build “gadgets” in the triangulation:
    • Variable gadgets force a choice that corresponds to setting a logical variable to True or False (you can’t pick both because of double blocking).
    • Clause gadgets check whether at least one variable in the clause is set to make the clause true (they let you include a certain piece if and only if the clause is satisfied).
    • The conflict graph of the whole construction mirrors the logic formula: the largest set of non-circular (acyclic) choices in the conflict graph corresponds to the maximum number of satisfiable clauses.
  • Upper and lower bounds that meet: Using the blow-ups, they prove:
    • An upper bound: given any big “non-circular” set in the conflict graph, you can follow the arrows and flip in about as many steps as predicted.
    • A matching lower bound: any flip sequence must at least pay that many steps.
    • Together, this pins down the flip distance closely in terms of the conflict graph, which is tied to the logic formula. That’s how NP-hardness is proved.

What they found and why it matters

  • Main result: It is NP-complete to decide whether the flip distance between two triangulations of a convex polygon is at most k. The same is true for the rotation distance between two binary trees.
  • Why this is big news: People have studied this problem since the 1980s. There were hints of “nice” behavior (for example, a “happy edge” never needs to be flipped if it appears in both the start and end triangulations), and the total number of flips between any two triangulations has a known linear upper bound. Still, the exact minimum number was unknown algorithmically. This paper settles the question: in general, there’s almost certainly no fast method that always finds the optimal number.
  • Wider ripple effects:
    • Associahedron: The “flip graph” (the network where triangulations are points and flips are edges) is the 1-skeleton of a famous polytope called the associahedron. The result shows it’s NP-hard to find shortest paths on that polytope’s edges.
    • Tamari lattice and relatives: Many mathematical structures related to triangulations and bracketings inherit this hardness result for finding shortest “cover” paths.
    • Binary trees: Since rotations are used in balancing trees (a staple in data structures), this tells us that computing the exact minimum number of rotations between two trees is also NP-complete—even though practical balancing uses simple, not necessarily minimal, rotation sequences.
    • Other Catalan structures: Even though many structures are counted by the same “Catalan numbers,” they behave differently computationally. For example, perfect matchings on points in convex position remain easy, while triangulations are hard—so no bijection that preserves the flip operation can make the triangulation problem easy.

What this could mean going forward

  • Algorithm design: Don’t expect a general, always-fast algorithm that computes the exact optimal flip/rotation distance. Instead, research can focus on:
    • Good approximations.
    • Fixed-parameter algorithms (fast when k is small).
    • Special cases that might be easier.
  • Mathematics and geometry: The result spreads to many geometric and combinatorial objects connected to triangulations, polytopes like the associahedron, and lattices like the Tamari lattice. It clarifies which shortest-path problems on these structures are fundamentally hard.
  • Computer science: For binary trees, the result gives a clean theoretical boundary between what’s possible in principle and what is efficient in practice. It explains why practical tree algorithms use simple local strategies rather than trying to find globally minimal rotations.

In short: The paper resolves a long-standing open problem by proving that finding the exact smallest number of flips (or rotations) is NP-complete, builds new tools (linear views, conflict graphs, blow-ups) to analyze flip sequences, and shows broad implications across geometry, algebra, and data structures.

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Knowledge Gaps

Unresolved gaps, limitations, and open questions

Below is a consolidated list of concrete knowledge gaps and open problems that remain after this work. Each item points to a direction where further research could make tangible progress.

  • Beyond NP-completeness: approximation complexity
    • Does flip/rotation distance on convex polygons/binary trees admit a polynomial-time constant-factor approximation or a PTAS? Is the problem APX-hard or hard to approximate within any constant/additive factor?
    • Can the reduction be turned into an L-reduction (or gap-preserving reduction) to transfer known inapproximability for Planar Separable Monotone Max-2SAT?
  • Parameterized complexity refinements
    • Tight FPT bounds in the parameter k (distance): can we achieve 2{O(k)} n{O(1)} with a small base or prove ETH/SETH lower bounds ruling out 2{o(k)} n{O(1)} algorithms?
    • Kernelization: does flip distance admit a polynomial kernel parameterized by k (or by other natural parameters, e.g., number of non-common diagonals)?
  • Structural and algorithmic theory of conflict graphs
    • Exact characterization: which directed graphs arise as conflict graphs of pairs of convex polygon triangulations (under linear representations)? Is there a polynomial-time recognition algorithm for this class?
    • Tractable subclasses: identify graph-structural restrictions (e.g., bounded pathwidth/treewidth along the spine, bounded maximum indegree/outdegree, interval/comparability-like constraints) under which maximum acyclic subset and the induced flip distance become polynomial-time solvable.
    • Approximability and FPT status of maximum acyclic subset restricted to conflict graphs (distinct from general digraphs), including parameterizations by solution size or structural measures.
  • Strengthening the reduction and bounds
    • Eliminate or reduce the blow-up: can NP-hardness be proved without large blow-ups, or with β subquadratic (or even O(1)) in n?
    • Tight bounds for all β: the current upper/lower bounds align asymptotically for large β but are loose for small β. Can we derive bounds with O(n) (or better) additive slack, or an exact formula for the flip distance in terms of conflict-graph features without requiring β to dominate?
    • Minimize constants: is β = 6(n2 + n) necessary, or can the threshold be lowered while preserving the reduction’s correctness?
  • Open reconfiguration problems on convex point sets
    • Almost-perfect matchings: the complexity of flip distance on convex point sets remains open. Is it in P, NP-complete, or FPT in natural parameters?
    • Non-crossing paths on point sets in general position: complexity of flip distance remains open; can NP-hardness (or a polynomial-time algorithm) be established?
  • Cosmohedra and permuto-associahedra: non-leaving-face property
    • To transfer NP-hardness to shortest paths on these polytopes’ 1-skeleta, it suffices to prove that shortest paths between vertices lying in the flip-graph subgraph never leave that subgraph (strong convexity/non-leaving-face property). This property is open for these polytopes; proving or refuting it would settle the complexity of shortest paths there.
  • Special-case algorithms for rotation distance
    • Targeted families: is rotation distance to specific target trees (e.g., balanced tree, comb/caterpillar, or trees with bounded imbalance) computable in polynomial time?
    • Large common-edge regime: when the number of shared diagonals is large (or when the conflict graph is acyclic), can the exact distance be computed in near-linear time via a direct formula?
  • Counting and structure of shortest sequences
    • Complexity of counting the number of shortest flip (rotation) sequences; is it #P-hard? Are there canonical normal forms or concise certificates of optimality beyond exhibiting a sequence of length k?
    • Can one efficiently compute or approximate a topologically constrained shortest sequence when partial order constraints (must-happen-before relations) are given?
  • Average-case and probabilistic analyses
    • What is the expected flip/rotation distance between random triangulations/binary trees on n vertices? Can it be approximated in subquadratic time?
    • Do random instances yield simpler conflict graphs (e.g., typically acyclic/bounded-width), enabling near-linear-time approximation or exact algorithms?
  • Delineating the boundary between “Catalan-hard” and “Catalan-easy”
    • Despite bijections, flip distance is NP-hard for triangulations and polynomial-time solvable for perfect matchings on convex point sets. What structural property (e.g., commutation of flips under bijection, conflict-graph tameness) predicts tractability across Catalan families?
    • Can this principle classify other Catalan structures by the complexity of their reconfiguration distance?
  • Generalized associahedra and related lattices/polytopes
    • For the many generalizations where NP-hardness now follows by corollary, what are the best possible approximation/FPT guarantees? Can we identify maximal subclasses (e.g., stellohedra-like cases) admitting polynomial-time distance computation?
  • Practical lower bounds and certification
    • Develop stronger, instance-specific lower bounds that do not rely on large β and can certify near-optimality of computed sequences in practice.
    • Efficiently computable surrogates for ac(H) that yield provable approximation guarantees on flip distance.
  • Variants and extensions
    • Weighted flip/rotation costs or constrained flips (e.g., forbid flipping certain diagonals): complexity and approximability status.
    • Unlabeled/partially labeled versions of rotation distance; effects of allowing double rotations or other local operations on complexity.
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Practical Applications

Immediate Applications

Below are practical, deployable applications that leverage the paper’s findings and techniques, organized by sector and use case. Each item includes key assumptions or dependencies that affect feasibility.

  • Software engineering (data structures and systems)
    • Distance-aware tree synchronization: Use conflict-graph-based upper/lower bounds to estimate the minimum number of rotations needed to transform one binary tree into another during state synchronization (e.g., distributed caches, replicated indices). Assumption: Rotation distance equals flip distance; exact minimal rotation computation is NP-hard, so systems should rely on bounds or heuristics rather than exact optima.
    • FPT-based maintenance for small-k scenarios: Integrate existing fixed-parameter tractable (FPT) algorithms for rotation/flip distance into tools that only need exact transformations when the distance k is small (e.g., in near-synchronized replicas). Dependency: Performance benefits only when k is small; worst-case is intractable.
    • Heuristic rebalancing diagnostics: Provide diagnostics that compare heuristic tree balancing (AVL, Red-Black, Treaps) against conflict-graph bounds to quantify how far from minimal a rebalancing sequence is. Assumption: Operational balancing rarely requires a globally minimal rotation sequence; bounds offer actionable insight without needing exact solutions.
  • Computational geometry and CAD/mesh processing
    • Triangulation alignment tools: For convex polygon triangulations (e.g., 2D meshing of convex shapes), offer flip-sequence planners that produce paths with guarantees based on conflict-graph upper/lower bounds instead of claiming optimality. Dependency: Convex polygon assumption; exact minimal flip sequences are NP-hard; conflict-graph construction must be feasible.
    • Quality-preserving flip workflows: Use acyclic-subset heuristics to prioritize flips that avoid conflicts, guiding mesh clean-up/reconfiguration while acknowledging intractability of exact minimal paths. Assumption: Works best when differences are localized; otherwise rely on heuristic selection and bounds.
  • Developer tools and libraries
    • Rotation/flip distance estimator library: A package that ingests two triangulations (linear representation) or binary trees and returns provable upper/lower bounds using the paper’s conflict graphs, acyclicity lemmas, and blow-up-based analyses. Dependency: Computing maximum acyclic subset is itself NP-hard; employ approximation/heuristic or FPT routines.
    • Visualization and certification: A tool that renders the linear representation, conflict graph, and proposed flip/rotation sequences, with certificates showing adherence to bounds and the happy-edge property in convex settings. Assumption: Convex case and proper mapping between trees and triangulations per Sleator–Tarjan–Thurston.
  • Academic research workflows
    • NP-hardness proof pattern reuse: Apply the linear representation, blow-ups, and conflict-graph framework to other reconfiguration graphs (paths, matchings, pseudo-triangulations) to either prove hardness or design parameterized algorithms. Dependency: Requires bijections/reductions that preserve local moves; matching and path variants have different flip definitions.
    • Bound-driven experimentation: Use the derived upper/lower bounds to benchmark heuristics for flip/rotation distances, enabling reproducible comparisons without exact solutions. Assumption: Benchmarks target convex or appropriately structured instances; general-position point sets differ.
  • Education and training
    • Course modules and assignments: Incorporate this resolved long-standing open problem (NP-completeness of rotation distance) into curricula on algorithms, computational geometry, and combinatorics; build exercises around constructing conflict graphs and bounding distances. Dependency: Students need background on Catalan structures, associahedra, and the tree–triangulation bijection.
  • Standards and policy (complexity-aware guidance)
    • Best-practice guidance: In standards for tree-based indices (databases, filesystems), explicitly discourage requirements for globally minimal rotation sequences, recommending heuristics and small-k exact methods. Assumption: Policymaking bodies accept complexity constraints; target systems where exact minimality is sometimes requested.

Long-Term Applications

These use cases will benefit from further research, scaling, or development—particularly approximation algorithms, specialized subclasses, and cross-domain integrations.

  • Approximation algorithms and guarantees
    • Approximation schemes for rotation/flip distance: Develop and deploy algorithms with provable approximation factors for binary-tree rotation distance and convex triangulation flip distance, guided by conflict-graph structure. Dependency: Need new approximation bounds; current exact computation is NP-hard.
    • Learning-guided reconfiguration: Use machine learning to predict near-minimal sequences in practical distributions of tree or triangulation changes, constrained by the conflict-graph topology. Assumption: Adequate training data; interpretability and safety checks via bounds.
  • Specialized algorithms for structured instances
    • Subclass-focused FPT improvements: Create faster FPT algorithms or polynomial-time methods for special families (e.g., near-balanced trees, constrained triangulations, bounded crossing structures), integrating them into database/index maintenance and geometry pipelines. Dependency: Structural assumptions must hold in the target systems.
  • Reconfiguration planning in robotics and modular systems
    • Skeleton-path planning on generalized associahedra: Model discrete configuration changes (e.g., modular robot reassembly) on associahedron-like graphs; use heuristics and bounds since shortest 1-skeleton paths are NP-hard in many generalizations. Dependency: Domain abstraction to Catalan-like reconfiguration graphs; acceptance that exact shortest paths may be intractable.
    • Hybrid continuous–discrete planners: Combine continuous motion planning with discrete lattice navigation (Tamari/Cambrian lattices), using bounds to manage discrete steps efficiently. Assumption: Non-leaving-face properties or convexity may be needed to keep paths within meaningful subgraphs.
  • Mathematical physics and combinatorics (polytope/lattice navigation)
    • Cluster algebra and scattering-amplitude tooling: As distances in generalized associahedra are NP-hard, design approximation or parameterized methods for navigating these structures in applications (e.g., amplitude computations, brick polytopes). Dependency: Domain-specific mappings; some polytopes (e.g., cosmohedra, permuto-associahedra) may require proof of strong convexity/non-leaving-face properties to transfer hardness cleanly.
    • Lattice-based optimization frameworks: For Tamari, Cambrian, Grid-Tamari, and related lattices, build solvers that approximate shortest cover-paths, incorporating greedy strategies validated by conflict-based bounds. Assumption: Users accept heuristic outputs with certificates of bounds rather than exact optima.
  • Data interoperability and diffing for hierarchical formats
    • Tree-structured diff tools: Develop diff/merge tools for ASTs, XML/JSON, and hierarchical schemas that use rotation-like operations and provide guaranteed bounds on edit distances via conflict graphs. Dependency: Mapping of edits to rotations is domain-specific; practical systems need robust approximate measures.
  • UI/UX for complexity-aware systems
    • “Distance meters” and progress indicators: In mesh editors and tree visualizers, expose distance bounds (not exact distances) to set user expectations, guide operations, and prevent misleading claims of optimality. Assumption: Users value transparent complexity-aware feedback.
  • Community resources and datasets
    • Benchmark suites and repositories: Curate instances (triangulations/trees, conflict graphs) with known bounds to evaluate algorithms and heuristics over time. Dependency: Community adoption; standardized input formats and evaluators.
  • Standards and best practices (long-term)
    • Complexity-informed specifications: Evolve standards to recognize NP-hardness of exact shortest reconfiguration paths in Catalan-structure-derived systems, recommending parameterized or approximate methods and bound certification. Dependency: Cross-stakeholder consensus; alignment with performance and reliability goals.
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Glossary

  • 1-skeleton: The graph formed by the vertices and edges of a polytope. "The $1$-skeleton of the Associahedron is exactly the flip graph of triangulations of a convex polygon with n+1n+1 vertices."
  • APX-hard: A complexity classification indicating no PTAS unless P=NP; optimization is hard to approximate within some constant factor. "the combinatorial distance problem is \NP-hard, and even \APX-hard, for polymatroids that are described by linarly many inequalities"
  • Associahedron: A convex polytope whose 1-skeleton encodes flip operations on Catalan structures. "The Associahedron KnK_n is an (n−2)(n-2)-dimensional convex polytope discovered by Stasheff"
  • β-blow-up: A construction that subdivides shared boundary edges of paired triangles and adds fans to amplify conflicts in flip sequences. "To construct the β\beta-blow-up of the triangulations TT and T′T', we apply the following procedure"
  • Brick polytopes: Polytopes associated with sorting networks and subword complexes, generalizing associahedra. "Brick polytopes~\cite{MR2820759,MR3327085}."
  • Cambrian Lattice: A lattice structure generalizing Tamari orders using orientations of Coxeter graphs. "The Type A Cambrian Lattice of Reading~\cite{reading2006cambrian}."
  • Catalan numbers: A classical sequence counting many combinatorial objects including triangulations and binary trees. "There are many combinatorial objects which can be counted by the well-known Catalan numbers Cn= 1n+1(2nn)C_n =~\frac{1}{n+1}\binom{2n}{n}"
  • Cluster algebras: Algebraic structures whose combinatorics yield generalized associahedra. "Generalized associahedra arising from cluster algebras~\cite{MR1941227,MR1887642}."
  • Conflict graph: A directed graph encoding precedence constraints between triangle-pair replacements in blow-ups. "The conflict graph H=H(T,T′)H=H(T,T') is a directed graph that has vertex set V(H)=ΓV(H)=\Gamma"
  • Cosmohedra: Polytopes whose 1-skeletons contain associahedral flip graphs as subgraphs. "such as cosmohedra~\cite{Arkani_Hamed_2025} or permuto-associahedra~\cite{reiner1994coxeter}."
  • Cover relation (in a lattice): The immediate precedence relation where one element covers another with no intermediate element. "One triangulation covers another triangulation if they differ by a single flip"
  • Dual graph: The graph whose vertices represent faces of a plane graph, with edges for face adjacencies. "It is obtained by taking the binary tree as a tree in the dual graph of the triangulation."
  • Empty five-hole: A set of five points in convex position with no other points inside; a forbidden configuration in certain algorithms. "with no empty five-holes~\cite{eppstein2010happy}."
  • Fan (of a triangle in a blow-up): The set of added edges from the triangle’s apex to the subdivision points on its base edge. "We call Λ(Δ)\Lambda(\Delta) the fan of Δ\Delta."
  • Flip distance: The minimum number of flips needed to transform one triangulation into another. "The flip distance between TT and~T′T' is the minimum ℓ\ell for which a flip sequence of length ℓ\ell from TT to~T′T' exists."
  • Flip graph: The graph whose vertices are triangulations and edges represent single flips. "The flip graph of triangulations of PP is the graph that has all triangulations of PP as vertices"
  • Flip sequence: A sequence of triangulations where consecutive ones differ by a single flip. "A flip sequence between an initial triangulation TT and a target triangulation~T′T' is a sequence T=T0,T1,…,Tℓ−1,Tℓ=T′T = T_0, T_1, \ldots, T_{\ell-1} , T_\ell = T' such that each TiT_i is a triangulations and two consecutive triangulations only differ by a single flip."
  • FPT-algorithm: A fixed-parameter tractable algorithm whose complexity is efficient for small parameter values. "The arguably most impressive work happened with regard to FPT-algorithms"
  • Generalized associahedra: A family of polytopes extending the associahedron, often arising from cluster algebras. "Generalized associahedra arising from cluster algebras~\cite{MR1941227,MR1887642}."
  • Generalized permutahedra: Polytopes obtained by deforming the permutahedron while preserving edge directions. "Generalized permutahedra~\cite{MR2487491}."
  • Graph associahedra: Polytopes associated with graphs that generalize the associahedron construction. "Those include graph associahedra and polymatroids~\cite{ito2023hardness}."
  • Grassmann-Tamari order: A Tamari-type partial order on objects related to Grassmannians. "The Grassmann-Tamari order of Santos, Stump, and Welker~\cite{santos2017noncrossing}."
  • Grid-Tamari Lattices: Lattices defined on certain grid-like combinatorial objects generalizing Tamari orders. "The Grid-Tamari Lattices of McConville~\cite{mcconville2017lattice}."
  • Happy edge property: A property where common edges between initial and target configurations remain unflipped in a shortest sequence. "All these hardness proofs are built around counterexamples to the happy edge property."
  • Hyperbolic geometry: Non-Euclidean geometry used in bounding flip graph diameters. "The proof relies on hyperbolic geometry and does not give a specific value"
  • Lattice (order theory): A partially ordered set where any two elements have a unique meet and join. "In general, a lattice is a partially ordered set with a join relation and a meet relation."
  • Linear representation: A drawing of a convex polygon’s triangulation using a horizontal spine and semicircular edges. "For our approach we use a different way to depict the triangulation, namely a linear representation as depicted in \Cref{fig:linear}."
  • Mutliassociahedron: A polytope generalizing the associahedron to higher “multi” settings. "The Mutliassociahedron~\cite{MR3221537,MR2167478}."
  • Non-leaving face property: The property that shortest paths between two vertices of a face stay within that face. "Analogs to this property are known as strong convexity or non-leaving face property and have been proven for similar structures"
  • Permuto-associahedra: Polytopes blending features of permutahedra and associahedra. "such as cosmohedra~\cite{Arkani_Hamed_2025} or permuto-associahedra~\cite{reiner1994coxeter}."
  • Permutree Lattice: A lattice structure defined on permutrees that interpolates sorting and bracketing orders. "The Permutree Lattice by Pilaud and Pons~\cite{pilaud2018permutrees}."
  • Planar Separable Monotone Max-2SAT: A constrained satisfiability problem with planar, bipartitioned incidence structure and monotone clauses. "To prove \NP-hardness we reduce from the NP-hard problem Planar Separable Monotone Max-2Sat~\cite{buchin2019geometric}."
  • Pointed pseudo-triangulation: A pseudo-triangulation where each vertex has at least one reflex angle. "A pseudo-triangulation is called pointed, if every vertex has one angle of at least $180$ degrees."
  • Polymatroid: A submodular polyhedral structure generalizing matroids, used in combinatorial optimization. "for polymatroids that are described by linarly many inequalities"
  • Pseudo-triangulation: A planar straight-line graph whose interior faces have exactly three convex angles. "A pseudo-triangulation is a collection of pair-wise crossing-free edges such that every inner face has exactly three convex angles."
  • Quotientopes: Polytopes arising from lattice quotients, generalizing permutahedra/associahedra frameworks. "Quotientopes ~\cite{MR3964495}."
  • Rotation (in a binary tree): A local restructuring that swaps a node with its child to modify tree shape. "A rotation in a binary tree is a node change that transforms one binary tree into another"
  • Rotation distance: The minimum number of rotations needed to transform one binary tree into another. "and therefore also computing the rotation distance of binary trees, is NP-hard."
  • Rotation graph: The graph whose vertices are binary trees and edges correspond to single rotations. "The rotation graph of binary trees on nn nodes is isomorphic to the flip graph of triangulations of convex (n+2n+2)-gons."
  • Spherical subword complexes: Combinatorial complexes associated with Coxeter groups; their dual graphs generalize associahedra. "Further, there are dual graphs of spherical subword complexes~\cite{MR3144391,MR2047852}"
  • Stellohedra: A subfamily of graph associahedra for which certain distance problems are polynomial-time solvable. "and a polynomial time algorithm for the special case of stellohedra~\cite{CARDINAL2024103877}."
  • Strong convexity (of a subgraph/face): A property ensuring shortest paths remain within a specified substructure. "Analogs to this property are known as strong convexity or non-leaving face property and have been proven for similar structures"
  • Tamari Lattice: A classic lattice on Catalan objects (e.g., triangulations, bracketings) ordered by rotations/flips. "The Tamari Lattice has been introduced by Tamari~\cite{tamari1962algebra}."
  • Topological ordering (of flips): An order of flips that respects dependency constraints so any respecting sequence is valid. "These results were followed up by algorithms that make use of a topological ordering of flips in triangulations."
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Authors (1)

  1. Joseph Dorfer 

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