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Arc Diagrams, Flip Distances, and Hamiltonian Triangulations

Authors Jean Cardinal, Michael Hoffmann, Vincent Kusters, Csaba D. Tóth, Manuel Wettstein

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Jean Cardinal
Michael Hoffmann
Vincent Kusters
Csaba D. Tóth
Manuel Wettstein

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Jean Cardinal, Michael Hoffmann, Vincent Kusters, Csaba D. Tóth, and Manuel Wettstein. Arc Diagrams, Flip Distances, and Hamiltonian Triangulations. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 197-210, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015) https://doi.org/10.4230/LIPIcs.STACS.2015.197

Abstract

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.

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Keywords
  • graph embeddings
  • edge flips
  • flip graph
  • separating triangles

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