Department of Computer Science | Institute of Theoretical Computer Science | CADMO

Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

Mittagsseminar (in cooperation with M. Ghaffari, A. Steger, D. Steurer and B. Sudakov)

Mittagsseminar Talk Information

Date and Time: Tuesday, October 01, 2019, 12:15 pm

Duration: 30 minutes

Location: CAB G51

Speaker: Emo Welzl

Bistellar and edge flip graphs of triangulations in the plane: Geometry and connectivity.

The set of all triangulations of a finite point set in the plane attains structure via flips: The graph, where two triangulations are adjacent if one can be obtained from the other by an elementary flip. This is an edge flip for full triangulations, or a bistellar flip for partial triangulations (where non-extreme points can be skipped). It is well-known (Lawson, 1972) that both, the edge flip graph and the bistellar flip graph are connected. For n the number of points and general position assumed, we show that the edge flip graph is (n/2-2)-connected and the bistellar flip graph is (n-3)-connected. Both bounds are tight. This matches the situation for regular triangulations (a subset of the partial triangulations), where (n-3)-connectivity was known through the secondary polytope (Gelfand, Kapranov, Zelevinsky, 1990) and Balinski’s Theorem. We show that the edge flip graph can be covered by 1-skeletons of polytopes of dimension at least n/2-2 (products of associahedra). Similarly, the bistellar flip graph can be covered by 1-skeletons of polytopes of dimension at least n-3 (products of secondary polytopes). (covers joint research with Uli Wagner, IST Austria)


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