## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

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

 Mittagsseminar Talk Information

Date and Time: Thursday, June 11, 2015, 12:15 pm

Duration: 30 minutes

Location: CAB G51

Speaker: Manuel Wettstein

## Vertical Visibility, Upward Triangulations, and 3-dimensional Catalan Numbers

We define the notion of a trapezoidal diagram of a plane graph G (embedded without crossings on a planar point set). Informally speaking, such a diagram is obtained by augmenting G with its trapezoidal decomposition and then forgetting about the exact locations of the vertices.

We study the number of such diagrams if the graphs G are either (a) perfect matchings or (b) triangulations. In both cases we give bijective proofs that establish certain relations with so called 3-dimensional Catalan numbers. We determine the exponential growth rates of the number of such diagrams as A^n and B^n, respectively, where n is the number of vertices, A = sqrt(27) = 5.196, and B = 27 / (729 * sqrt(3) / (40 * pi) - 9)^3 = 23.459.

Coincidentally, the number of such diagrams in the case of perfect matchings is equal to the number of different ways (in terms of the vertical visibility structure) that we can arrange n/2 non-intersecting convex shapes in the plane. In the case of triangulations, these diagrams are closely related to upward triangulations (i.e., directed maximal planar graphs that allow a plane embedding in which all edges are drawn as y-monotone curves pointing upwards).

Previous talks by year:   2018  2017  2016  2015  2014  2013  2012  2011  2010  2009  2008  2007  2006  2005  2004  2003  2002  2001  2000  1999  1998  1997  1996

Information for students and suggested topics for student talks