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.
@InProceedings{cardinal_et_al:LIPIcs.STACS.2015.197, author = {Cardinal, Jean and Hoffmann, Michael and Kusters, Vincent and T\'{o}th, Csaba D. and Wettstein, Manuel}, title = {{Arc Diagrams, Flip Distances, and Hamiltonian Triangulations}}, booktitle = {32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)}, pages = {197--210}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-78-1}, ISSN = {1868-8969}, year = {2015}, volume = {30}, editor = {Mayr, Ernst W. and Ollinger, Nicolas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.197}, URN = {urn:nbn:de:0030-drops-49141}, doi = {10.4230/LIPIcs.STACS.2015.197}, annote = {Keywords: graph embeddings, edge flips, flip graph, separating triangles} }
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