Simultaneous diagonal flips in plane triangulations

Authors:
Prosenjit Bose;Jurek Czyzowicz;Zhicheng Gao;Pat Morin;David R. Wood
Affiliations:
Carleton University, Ottawa, Canada;Université du Québec en Outaouais, Gatineau, Canada;Carleton University, Ottawa, Canada;Carleton University, Ottawa, Canada;Universitat Politècnica de Catalunya, Barcelona, Spain
Venue:
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Year:
2006

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Abstract

Simultaneous diagonal flips in plane triangulations are investigated. It is proved that every triangulation with at least six vertices has a simultaneous flip into a 4-connected triangulation, and that it can be computed in linear time. It follows that every triangulation has a simultaneous flip into a Hamiltonian triangulation. This result is used to prove that for any two n-vertex triangulations, there exists a sequence of O(log n) simultaneous flips to transform one into the other. The total number of edges flipped in this sequence is O(n). The maximum size of a simultaneous flip is then studied. It is proved that every triangulation has a simultaneous flip of at least 1/3(n − 2) edges. On the other hand, every simultaneous flip has at most n − 2 edges, and there exist triangulations with a maximum simultaneous flip of 6/7 (n − 2) edges.