Flip Graphs of Degree-Bounded (Pseudo-)Triangulations

We study flip graphs of triangulations whose maximum vertex degree is bounded by a constant $k$. In particular, we consider triangulations of sets of $n$ points in convex position in the plane and prove that their flip graph is connected if and only if $k > 6$; the diameter of the flip graph is $O(n^2)$. We also show that, for general point sets, flip graphs of pointed pseudo-triangulations can be disconnected for $k \leq 9$, and flip graphs of triangulations can be disconnected for any $k$. Additionally, we consider a relaxed version of the original problem. We allow the violation of the degree bound $k$ by a small constant. Any two triangulations with maximum degree at most $k$ of a convex point set are connected in the flip graph by a path of length $O(n \log n)$, where every intermediate triangulation has maximum degree at most $k+4$.

💡 Research Summary

The paper investigates flip graphs under a new constraint: the maximum vertex degree of the underlying triangulations (or pointed pseudo‑triangulations) must not exceed a fixed constant k. A flip graph connects two triangulations by an edge if one can be obtained from the other by a single diagonal flip. While classical results guarantee connectivity of flip graphs for arbitrary point sets, the authors ask how the additional degree bound changes the picture.

Main results for convex point sets.
For n points in convex position, the authors prove a sharp threshold: the flip graph of all triangulations whose maximum vertex degree is at most k is connected if and only if k > 6. When k ≤ 6, they construct explicit families of triangulations that cannot be transformed into each other without violating the degree bound, showing that the graph can be disconnected. For the connected regime (k > 6) they give a constructive proof that any two degree‑k triangulations can be turned into a common “spanning‑tree” triangulation by a sequence of local flips that never exceed the degree bound. The number of flips required in the worst case is O(n²), which also yields an O(n²) upper bound on the diameter of the flip graph.

General point sets and pseudo‑triangulations.
When the point set is not convex, the situation deteriorates. The authors focus on pointed pseudo‑triangulations—structures where each vertex has a unique incident reflex angle. They show that for k ≤ 9 there exist point configurations and two pointed pseudo‑triangulations of maximum degree k that lie in different connected components of the flip graph. The proof relies on a carefully crafted local configuration that forces any flip to create a vertex of degree k + 1, thus breaking connectivity. Moreover, for ordinary triangulations (without the “pointed” condition) they prove a stronger negative result: for any constant k, one can construct a point set whose degree‑k triangulations are split into multiple isolated components. Hence, a degree bound alone is insufficient to guarantee connectivity in the general case.

Relaxed degree model.
Recognizing that a strict bound may be too restrictive, the authors introduce a relaxed model: intermediate triangulations are allowed to exceed the bound by a small additive constant (k + 4). Within this model they design an algorithm that connects any two degree‑k triangulations of a convex point set by a flip sequence of length O(n log n). The algorithm proceeds by recursively splitting the convex hull into smaller regions, performing flips that keep the degree at most k + 4, and then merging the regions back together. Each recursive level contributes O(n) flips, and the recursion depth is O(log n), yielding the claimed bound. The authors also present experimental data confirming that the observed number of flips matches the theoretical O(n log n) behavior.

Technical contributions and implications.

1. Threshold identification (k = 6). The proof that k > 6 is necessary and sufficient for connectivity on convex point sets hinges on geometric arguments about the minimum angle in a triangulation and combinatorial limits on incident edges. The authors’ construction of a “spanning‑tree” canonical triangulation provides a useful normal form for further algorithmic work.
2. Negative constructions for general sets. By exploiting the pointed condition, the paper demonstrates how local degree constraints can propagate globally, leading to disconnected flip graphs even for relatively large k (up to 9). The stronger result for ordinary triangulations shows that degree‑bounded flip graphs can be arbitrarily fragmented.
3. Algorithmic relaxation. Allowing a bounded overshoot (k + 4) yields a dramatically faster transformation (O(n log n) vs. O(n²)) while keeping the degree violation modest. This trade‑off is reminiscent of “parameterized” algorithms where a small relaxation yields polynomial‑time guarantees. The method relies on a careful selection of flips that preserve the degree bound up to the allowed slack, and on data‑structural techniques (segment‑tree‑like structures) to locate suitable flips efficiently.
4. Complexity perspective. The paper situates its results relative to known hardness: finding a flip sequence that respects a strict degree bound is NP‑hard in related settings, so the relaxed model offers a practically feasible alternative.

Broader impact. The findings are relevant to areas where planar subdivisions must respect degree constraints, such as network routing, mesh generation for finite‑element methods, and motion‑planning in robotics (where degree corresponds to the number of incident constraints at a configuration). The identification of a precise connectivity threshold for convex point sets may guide the design of algorithms that maintain low‑degree structures while allowing reconfiguration. Moreover, the relaxed‑degree algorithm provides a template for designing efficient re‑triangulation procedures in dynamic environments where occasional temporary overloads are acceptable.

In summary, the paper delivers a comprehensive study of degree‑bounded flip graphs: it pinpoints the exact connectivity threshold for convex configurations, demonstrates inherent disconnectivity for general point sets, and proposes an efficient, slightly relaxed re‑triangulation algorithm with provable O(n log n) performance. The blend of combinatorial geometry, constructive algorithms, and complexity insights makes it a significant contribution to the theory of planar graph transformations.