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

Prof. Emo Welzl and Prof. Bernd Gärtner

Mittagsseminar Talk Information |

**Date and Time**: Tuesday, January 29, 2013, 12:15 pm

**Duration**: 30 minutes

**Location**: OAT S15/S16/S17

**Speaker**: Emo Welzl

An embedding of a graph G in the plane is called an upward drawing, if every edge is represented by a y-monotone Jordan curve, and it is called plane, if no two such curves meet other than at common endpoints. Such an upward drawing induces an orientation of the edges of G (where the notion ”upward” even indicates which direction we want to go) – this gives a directed graph G. We discuss the problem of determining the maximum and the minimum number of upward planar orientations a maximal planar graph can have (and we even indicate where the question came from, even though this turned out to be a dead end). We show that every maximal planar graph (triangulation) has at most O(4^{n} poly(n)) upward planar orientations, n the number of vertices of the graph. (Other bounds to be mentioned are that every maximal planar graph has at least Omega(1.189^{n}) upward planar orientations and that there exist maximal planar graphs having as few as O(2^{n}) upward planar orientations and maximal planar graphs having as many as Omega(2.599^{n}) upward planar orientations - from joint work with Fabrizio Frati and Joachim Gudmundsson.)

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