A History of Flips in Combinatorial Triangulations

@inproceedings{Bose2011AHO,
  title={A History of Flips in Combinatorial Triangulations},
  author={Prosenjit Bose and Sander Verdonschot},
  booktitle={EGC},
  year={2011}
}
Given two combinatorial triangulations, how many edge flips are necessary and sufficient to convert one into the other? This question has occupied researchers for over 75 years. We provide a comprehensive survey, including full proofs, of the various attempts to answer it. 
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18 Citations
Background Citations
8

18 Citations

Flipping in Spirals
A Lower Bound on the Diameter of the Flip Graph
TLDR
The diameter of the flip graph is shown to be at least $\frac{7n}{3} + \Theta(1)$, improving upon the previous $2n + Theta (1)$ lower bound.
Modular flip-graphs of one-holed surfaces
Making triangulations 4-connected using flips
Transforming plane triangulations by simultaneous diagonal flips
Flipping edge-labelled triangulations
Edge Flips in Combinatorial Triangulations
TLDR
This project will investigate a topic that has fascinated geometers for many years: the flip graph of triangulations, which is considered abstract unlabeled (i.e. isomorphism classes of) triangulation.
Construction of acyclically 4-colourable planar triangulations with minimum degree 4
TLDR
This paper describes a method for constructing all 4-connected acyclically 4-colourable planar triangulations that have exactly four odd-vertices, except the ones that contain no adjacent odd-vertedices.
Arc diagrams, flip distances, and Hamiltonian triangulations
Flip Distances Between Graph Orientations
TLDR
It is proved that deciding whether the flip distance between two $\alpha$-orientations of a planar graph $G$ is at most two is \NP-complete, and this also holds in the special case of perfect matchings, where flips involve alternating cycles.
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References

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Triangulations without pointed spanning trees
THE DIAGONAL FLIPS OF TRIANGULATIONS ON THE SPHERE
. It will be shown that any two triangulations with $n$ vertices on the sphere can be transformed into each other by at most $8n-54$ diagonal flips if $n\geq 13$ and $8n-48$ if $n\geq 7$ .
Diagonal Flips in Labelled Planar Triangulations
TLDR
It is shown that any two labelled triangulations can be transformed into each other using at most O(nlogn) diagonal flips and it is proved that if the minimum degree of a triangulation is at least 4, then it contains at least 2n + 3 flippable edges.
Diagonal Flips in Hamiltonian Triangulations on the Sphere
TLDR
It is proved that any two Hamiltonian triangulations on the sphere with n≥5 vertices can be transformed into each other by at most 4n−20 diagonal flips, preserving the existence of Hamilton cycles.
Making triangulations 4-connected using flips
Flips in planar graphs
Short Encodings of Evolving Structures
A derivation in a transformational system such as a graph grammar may be redundant in the sense that the exact order of the transformations may not affect the final outcome; all that matters is that…
Bemerkungen zum Vierfarbenproblem.
Diagonal Transformations of Graphs on Closed Surfaces
A Theorem on Graphs