=== When ===

Tuesday May 19th at 12:00

=== Where ===

Department of Engineering

Section of Computer Science and Automation

Via della Vasca Navale, 79

Meeting room (1.10) on 1st floor

=== Title ===

Arc diagrams, flip distances, and Hamiltonian triangulations

=== Speaker ===

Dr. Michael Hoffmann, ETH Zürich

http://www.inf.ethz.ch/personal/hoffmann/

=== Abstract ===

A flip is a local operation on a triangulation (maximal planar graph) that switches the diagonal of a 4-cycle. The flip graph is a graph whose vertices are all triangulations on n vertices and two triangulations are adjacent if they differ by exactly one flip. These graphs have been studied in various settings and have proven to be a very useful and versatile tool to analyze the structure of triangulations, both combinatorially and algorithmically.

In this talk I will present recently improved bounds concerning the combinatorial flip graph, where triangulations are considered to be abstract graphs (rather than, for instance, geometric realizations on a concrete point set). Specifically we show that from every triangulation on n>=6 vertices we can reach a Hamiltonian triangulation by a sequence of less than n/2 flips. An interesting bit about this bound is that it was known that 3n/5-c flips are always sufficient and sometimes necessary to reach a 4-connected triangulation (which by Tutte’s/Whitney’s Theorem is Hamiltonian). As a byproduct we improve the upper bound on the diameter of the flip graph from ~5.2n to ~5n. Finally, I will discuss an application of our result to a graph drawing problem, showing 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.

(joint work with Jean Cardinal, Vincent Kusters, Csaba D. Tóth, and Manuel Wettstein)