Arc Diagrams, Flip Distances, and Hamiltonian Triangulations
We show that every triangulation (maximal planar graph) on n\ge 6 vertices can be flipped into a Hamiltonian triangulation using a sequence of less than n/2 combinatorial edge flips. The previously best upper bound uses 4-connectivity as a means to establish Hamiltonicity. But in general about 3n/5 flips are necessary to reach a 4-connected triangulation. Our result improves the upper bound on the diameter of the flip graph of combinatorial triangulations on n vertices from 5.2n-33.6 to 5n-23. We also show that for every triangulation on n vertices there is a simultaneous flip of less than 2n/3 edges to a 4-connected triangulation. The bound on the number of edges is tight, up to an additive constant. As another application we show that every planar graph on n vertices admits an arc diagram with less than n/2 biarcs, that is, after subdividing less than n/2 (of potentially 3n-6) edges the resulting graph admits a 2-page book embedding.
graph embeddings
edge flips
flip graph
separating triangles
197-210
Regular Paper
Jean
Cardinal
Jean Cardinal
Michael
Hoffmann
Michael Hoffmann
Vincent
Kusters
Vincent Kusters
Csaba D.
Tóth
Csaba D. Tóth
Manuel
Wettstein
Manuel Wettstein
10.4230/LIPIcs.STACS.2015.197
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