Compatible Flip Graphs: Theory & Applications

• Compatible flip graphs are reconfiguration graphs defined on combinatorial and geometric objects where flips must meet specific non-crossing or foldability conditions.
• They have been explored in settings such as non-crossing spanning trees, triangulations with forbidden edges, and Miura-ori origami, linking theory with practical geometric applications.
• Key findings include NP-completeness of flip sequence optimization, refined diameter bounds, and detailed connectivity characterizations across diverse models.

Searching arXiv for papers on compatible flip graphs and related flip-graph variants. A compatible flip graph is a restricted flip graph in which vertices are admissible combinatorial or geometric objects and adjacency is defined only by flips that preserve an additional compatibility condition. In the setting of non-crossing spanning trees on planar point sets, a compatible flip is an edge exchange $T' = T-e+f$ in which the removed edge and the added edge do not cross; in the $m\times n$ Miura-ori, compatibility is realized by face flips between flat-foldable mountain–valley assignments, equivalently by single-vertex recolorings in a proper 3-coloring reconfiguration graph modulo global color rotation (Bjerkevik et al., 23 Mar 2026, Gupta, 21 Jun 2026). This suggests that the term is best understood as a family of restricted reconfiguration graphs rather than a single canonical object.

1. Restricted flip systems and the meaning of compatibility

For non-crossing spanning trees on a point set $P$ in general position, the ambient flip graph has one vertex for each non-crossing spanning tree on $P$, and an edge between two trees whenever one can be obtained from the other by exchanging a single edge while preserving both the tree property and non-crossing. Within this framework, a compatible flip is the special case in which the exchanged edges themselves are compatible, meaning that the interiors of the removed and added segments are disjoint. The same paper also studies rotations, in which the removed and added edges share an endpoint. Avis and Fukuda showed that, for point sets in general position, the flip graph of non-crossing spanning trees is connected for flips, compatible flips, and rotations (Bjerkevik et al., 23 Mar 2026).

For the Miura-ori crease pattern, the origami flip graph $OFG(C)$ has as vertices all locally valid mountain–valley assignments, and adjacency is given by a face flip that switches the assignment of every crease bordering a single face while remaining locally valid. In the $m\times n$ Miura-ori, the paper writes $OFG(m,n)$ for this graph, and uses the known bijection

$$OFG(m,n)\;\cong\; R_3(G_{m,n}) /(\mathbb{Z}/3\mathbb{Z}),$$

where $R_3(G_{m,n})$ is the 3-coloring reconfiguration graph of the grid graph $G_{m,n}$. In this model, compatibility is exactly the condition that a single-vertex recoloring remain proper, or equivalently that the corresponding face flip preserve flat-foldability (Gupta, 21 Jun 2026).

A related constrained viewpoint arises for triangulations of planar point sets when certain geometric edges are forbidden. If $m\times n$0 is a set of forbidden edges, $m\times n$1 denotes the flip graph induced by triangulations containing none of the edges in $m\times n$2. Here compatibility is with the forbiddance condition rather than with a local exchange rule: only triangulations avoiding $m\times n$3, and flips that never introduce $m\times n$4, remain in the state space (Bigdeli et al., 2022).

2. Compatible flips for non-crossing spanning trees

Let $m\times n$5 be a set of $m\times n$6 points in the plane in general position. A non-crossing spanning tree on $m\times n$7 is a straight-line tree $m\times n$8 with $m\times n$9, connected and acyclic, and with edges pairwise non-crossing. A standard flip is an operation $P$0 where $P$1, $P$2 is a segment between two points of $P$3 not already in $P$4, and $P$5 is again a non-crossing spanning tree. A compatible flip is the refinement in which the exchanged edges $P$6 and $P$7 do not cross in the geometric drawing. Rotations form an even more restricted class: $P$8 with $P$9, $P$0, and hence a shared endpoint $P$1 (Bjerkevik et al., 23 Mar 2026).

The paper formalizes the corresponding restricted flip graph by taking all non-crossing spanning trees as vertices and joining two trees if they differ by a compatible flip. The resulting graph is not assigned a separate standard name in the literature of that paper, but it is explicitly described as the flip graph where edges correspond to compatible flips rather than arbitrary edge exchanges. The same treatment applies to the rotation flip graph (Bjerkevik et al., 23 Mar 2026).

The principal geometric regime is the convex case, where the points lie on a circle and are linearly represented by cutting the circle open into a spine with vertices $P$2. In that representation, edges of one tree are drawn as semicircles above the spine and edges of the other tree below it. This linear model supports the definition of edge length, coverage, gaps, and the gap-edge bijection, all of which are used to analyze compatible flip sequences and their obstructions (Bjerkevik et al., 23 Mar 2026).

3. Conflict graphs, shortest paths, and diameter bounds

The central combinatorial tool for compatible flip graphs of non-crossing spanning trees is the conflict graph $P$3. Its vertices are the gaps corresponding to near-near pairs in the linear representation, and its directed edges encode flip dependencies: there is a directed edge from one gap to another when the edge currently present in the first pair must be removed before the flip associated with the second pair can be performed. The paper distinguishes three conflict types, involving crossings and coverage relations among the paired edges. A key parameter is

$P$4

the size of a largest acyclic induced subgraph (Bjerkevik et al., 23 Mar 2026).

Theorem 2.1, restated from the earlier BKUV framework, shows that the ratio $P$5 simultaneously controls upper and lower bounds. If $P$6, then

$P$7

and there is also a constant $P$8 such that

$P$9

Moreover, the sets $OFG(C)$0, $OFG(C)$1, and $OFG(C)$2 of above, below, and crossing near-near gaps are each acyclic sets of $OFG(C)$3, so the conflict graph isolates cycle structure that is invisible in a purely local description of flips (Bjerkevik et al., 23 Mar 2026).

The main complexity theorem states that the decision problem “is there a flip sequence of length at most $OFG(C)$4 between $OFG(C)$5 and $OFG(C)$6?” is NP-complete even for point sets in convex position. The restriction to compatible flips or to rotations does not reduce this complexity: the corresponding shortest-path problems remain NP-complete, again even in convex position. The reduction proceeds through the hardness of deciding whether $OFG(C)$7 for a conflict graph arising from a pair of non-crossing trees, together with blowup constructions that amplify the contribution of $OFG(C)$8 to the flip distance (Bjerkevik et al., 23 Mar 2026).

The same paper also refines diameter estimates. For $OFG(C)$9 points in convex position, the then-current best bounds were

$m\times n$0

It proves that if one of the trees is stacked, then the lower-bound construction based on a stacked tree is optimal up to a constant term: there exists a flip sequence of length at most $m\times n$1 to any other tree. It also improves the lower bound on the diameter to

$m\times n$2

Thus compatibility constraints do not make the geodesic problem simpler, and in the convex case they remain tightly coupled to the structure of the conflict graph (Bjerkevik et al., 23 Mar 2026).

4. Origami compatible flip graphs and height functions

For the $m\times n$3 Miura-ori, compatibility is encoded by a height-function model. Every proper 3-coloring $m\times n$4 of the grid graph $m\times n$5 lifts to a unique integer-valued height function

$m\times n$6

satisfying $m\times n$7 across every grid edge, and the color at a vertex is $m\times n$8. A face flip becomes a recoloring of one grid vertex, and it is allowed exactly when the recoloring remains proper. In height-function language, a face is flippable precisely when the corresponding grid vertex is a strict local extremum. For $m\times n$9,

$OFG(m,n)$0

This gives a purely discrete characterization of compatibility in terms of local extrema of a 1-Lipschitz function with unit steps (Gupta, 21 Jun 2026).

The same framework supports explicit degree counts. For $OFG(m,n)$1, $OFG(m,n)$2 has exactly four vertices of degree $OFG(m,n)$3, namely the four corner gradients

$OFG(m,n)$4

The number of degree-3 vertices is $OFG(m,n)$5. For $OFG(m,n)$6, the number of degree-4 vertices is

$OFG(m,n)$7

and for $OFG(m,n)$8, the number of degree-5 vertices is

$OFG(m,n)$9

The diameter problem is also transferred to the height-function setting: the distance between two states is given by an $$OFG(m,n)\;\cong\; R_3(G_{m,n}) /(\mathbb{Z}/3\mathbb{Z}),$$0-type dispersion of the height difference, minimized over global offsets, and the paper states that a closed-form lower bound for the diameter holds for all $$OFG(m,n)\;\cong\; R_3(G_{m,n}) /(\mathbb{Z}/3\mathbb{Z}),$$1 and $$OFG(m,n)\;\cong\; R_3(G_{m,n}) /(\mathbb{Z}/3\mathbb{Z}),$$2, while the matching upper bound reduces to an extremal inequality for 1-Lipschitz functions on the grid (Gupta, 21 Jun 2026).

The specialized $$OFG(m,n)\;\cong\; R_3(G_{m,n}) /(\mathbb{Z}/3\mathbb{Z}),$$3 Miura-ori admits sharper exact formulas. The origami flip graph $$OFG(m,n)\;\cong\; R_3(G_{m,n}) /(\mathbb{Z}/3\mathbb{Z}),$$4 has

$$OFG(m,n)\;\cong\; R_3(G_{m,n}) /(\mathbb{Z}/3\mathbb{Z}),$$5

vertices and

$$OFG(m,n)\;\cong\; R_3(G_{m,n}) /(\mathbb{Z}/3\mathbb{Z}),$$6

edges. Its minimum degree is $$OFG(m,n)\;\cong\; R_3(G_{m,n}) /(\mathbb{Z}/3\mathbb{Z}),$$7, and for all $$OFG(m,n)\;\cong\; R_3(G_{m,n}) /(\mathbb{Z}/3\mathbb{Z}),$$8 that minimum occurs exactly four times. Its maximum degree is $$OFG(m,n)\;\cong\; R_3(G_{m,n}) /(\mathbb{Z}/3\mathbb{Z}),$$9, attained exactly twice, and no vertex has degree $R_3(G_{m,n})$0. The degree set is

$R_3(G_{m,n})$1

The paper then proves that the diameter of $R_3(G_{m,n})$2 is

$R_3(G_{m,n})$3

using techniques from 3-coloring reconfiguration graphs (Christensen et al., 24 Jun 2025).

5. Connectivity, components, and constrained subspaces

Compatible flip graphs are not uniformly connected. In the triangulation setting with forbidden edges, a set $R_3(G_{m,n})$4 of edges is a flip cut set if $R_3(G_{m,n})$5 is disconnected, and a single forbidden edge $R_3(G_{m,n})$6 is a flip cut edge precisely when a crossing-edge graph $R_3(G_{m,n})$7 is disconnected; equivalently, it suffices to check the refined graph $R_3(G_{m,n})$8 built from empty convex quadrilaterals. This yields an $R_3(G_{m,n})$9 algorithm to test whether an edge is a flip cut edge and, after preprocessing, an $G_{m,n}$0 algorithm to test whether two triangulations lie in the same connected component of $G_{m,n}$1. For $G_{m,n}$2 points in convex position, the flip cut number is exactly $G_{m,n}$3 (Bigdeli et al., 2022).

For plane spanning paths on a point set $G_{m,n}$4, the connectivity of the full flip graph has remained open for more than 16 years, but several compatible subgraphs are now understood. For a fixed endpoint $G_{m,n}$5, the flip graph of $G_{m,n}$6 is connected when $G_{m,n}$7 has at most two convex layers. More generally, suffix-independent paths induce a connected subgraph, and the paper states that to answer the open problem affirmatively it suffices to show that each path can be flipped to some suffix-independent path. In convex position with one fixed endpoint, the flip graph of $G_{m,n}$8 has diameter $G_{m,n}$9 and radius $m\times n$00, with the spirals as exactly the centers (Kleist et al., 2024).

Other geometric models display both connectivity and failure of connectivity under compatibility constraints. For arrangements of pseudolines, the triangle-flip graph $m\times n$01 is $m\times n$02-connected for $m\times n$03 and $m\times n$04, and its connectivity equals its minimum degree, which is exactly $m\times n$05. For intersecting arrangements of pseudocircles, and also for cylindrical intersecting arrangements, triangle flips induce a connected flip graph; in both pseudoline and pseudocircle settings the diameter is in $m\times n$06 (Radtke et al., 2023). By contrast, for pseudo-triangulations with face degree at most $m\times n$07, the flip graph of $m\times n$08-DPTs is generally not connected, although the connected components can be computed; within a fixed tail class $m\times n$09, however, the induced subgraph is connected (Löffler et al., 2024).

Surface geometry produces further connected compatible flip graphs. For geometric triangulations with fixed vertices on a flat torus or on a closed hyperbolic surface, the flip graph is connected, and repeated Delaunay flips transform any geometric triangulation into a Delaunay triangulation. The paper also gives upper bounds on the number of edge flips needed to reach a Delaunay triangulation: at most $m\times n$10 on a flat torus and at most $m\times n$11 on a closed hyperbolic surface (Despré et al., 2019).

6. Broader compatible frameworks and recurrent themes

In higher-dimensional topology, the vertex-preserving flip graph $m\times n$12 of triangulated $m\times n$13 uses only $m\times n$14–$m\times n$15 and $m\times n$16–$m\times n$17 bistellar moves and excludes $m\times n$18–$m\times n$19 and $m\times n$20–$m\times n$21 moves because they change the vertex number. Within this fixed-$m\times n$22 compatible subgraph, $m\times n$23 and $m\times n$24 are connected, while isolated vertices are known in $m\times n$25, $m\times n$26, and $m\times n$27. The paper defines the polytopal closure $m\times n$28 as the connected component containing all boundary complexes of convex 4-polytopes, and proposes the Weeping Willow Conjecture to describe how non-polytopal components arise from the polytopal closure at larger vertex numbers (Faber et al., 10 May 2025).

Several adjacent literatures isolate compatible pieces inside larger flip graphs. In the flip graph on perfect matchings of $m\times n$29, the subgraph induced by type $m\times n$30 is isomorphic to the signed reversal graph $m\times n$31, and for a general type $m\times n$32, each connected component is isomorphic to

$m\times n$33

This gives a decomposition into compatibility classes relative to a fixed matching $m\times n$34 (Cioabă et al., 2020). For graph orientations, the dual problem of $m\times n$35-orientations becomes tractable under the compatible restriction that flips only change sinks into sources, or vice versa: the flip graph is then the cover graph of a distributive lattice, and a shortest vertex flip sequence can be computed in polynomial time (Aichholzer et al., 2019).

Compatibility also appears through duality and rigidity. For marked surfaces, the flip graph embeds in the arc complex as its dual, and finite rigidity of the flip graph implies finite rigidity of the arc complex. The resulting theorem applies to a broad class of surfaces and notably includes surfaces with boundary (Sadanand et al., 2023). In a different direction, quadrilateral flips on graphs preserve the face vector of the associated graphical zonotope: if one graph is obtained from another by such flips, then the graphical zonotopes have the same face vector, and all triangulations of the $m\times n$36-gon therefore yield graphical zonotopes with a common face vector (Xu, 2018).

A recurrent theme is that compatibility is neither synonymous with connectivity nor with algorithmic ease. Compatible flips and rotations for non-crossing spanning trees remain NP-hard in the shortest-path problem (Bjerkevik et al., 23 Mar 2026), $m\times n$37-dart pseudo-triangulation flip graphs are generally not connected (Löffler et al., 2024), and fixed-vertex bistellar compatibility in triangulated $m\times n$38 can produce isolated components (Faber et al., 10 May 2025). At the same time, many natural compatible models remain connected and admit strong structural descriptions, including Miura-ori origami graphs (Gupta, 21 Jun 2026), geometric triangulations on flat tori and closed hyperbolic surfaces (Despré et al., 2019), intersecting pseudocircle arrangements (Radtke et al., 2023), and plane spanning paths on point sets with at most two convex layers (Kleist et al., 2024).