## 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: Tuesday, May 11, 2010, 12:15 pm

Duration: This information is not available in the database

Location: CAB G51

Speaker: Tobias Müller (CWI Amsterdam)

## Line arrangements and geometric representations of graphs

A graph is a disk graph if it is the intersection graph of disks in the plane. That is, we can represent each vertex by a disk in such a way that two vertices are adjacent if and only if the corresponding disks intersect. Similarly one can define a segment graph as the intersection graph of line segments and d-ball graph as the intersection graph of balls in d-dimensional space. When d=2 this of course yields disk graphs, and when d=1 twe get the familiar well-studied class of interval graphs. A d-dot product graph is a graph that can be represented by vectors in d-space such that two vertices are adjacent if and only if the inner product of the corresponding vectors is at least one. When d=1 this yields the familiar well-studied class of threshold graphs.

By making use of some classical results in algebraic geometry, on pseudoline arrangements (also known as oriented matroids of rank 3) and on the Colin de Verdiere parameter, I will settle some conjectures and questions on these graph classes by Breu and Kirkpatrick, Fiduccia et al., Spinrad and (Van Leeuwen)^2 and improve upon a result by Kratochvil & Matousek.

Joint work with Colin McDiarmid and Ross Kang.

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