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

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

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Emo Welzl

Upward Planar Orientations of Maximal Planar Graphs

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(4ⁿ 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(1.189ⁿ) upward planar orientations and that there exist maximal planar graphs having as few as O(2ⁿ) upward planar orientations and maximal planar graphs having as many as Omega(2.599ⁿ) upward planar orientations - from joint work with Fabrizio Frati and Joachim Gudmundsson.)

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