## 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, November 01, 2011, 12:15 pm

Duration: 30 minutes

Location: CAB G51

Speaker: Vincent Kusters

## Characterizing graphs with a sliceable rectangular dual

A rectangular partition of a rectangle r is a partition of r into a set R of non-overlapping rectangles such that no four rectangles in R meet at one common point. A rectangular dual of a planar graph G is a rectangular partition R, such that (i) there is a one-to-one correspondence between the rectangles in R and the vertices of G, and (ii) two rectangles in R share a common boundary if and only if the corresponding vertices of G are connected. A given graph can have many rectangular duals. A rectangular dual is sliceable if it can be recursively subdivided along horizontal or vertical lines. We aim to characterize the graphs which have a sliceable rectangular dual.
If a graph G has a rectangular dual and does not have a separating 4-cycle, then G has a sliceable rectangular dual. No characterization of the sliceable subset of graphs with separating 4-cycles is known. We describe a new class of graphs with exactly one separating 4-cycle which is adjacent to the outer cycle and show that all graphs in this class are sliceable. In addition, we present a generic method for proving the sliceability of such graphs. We introduce rotating windmills and prove that they are nonsliceable. Finally, we perform experiments which lead us to believe that rotating windmills are the only nonsliceable graphs which have a rectangular dual and exactly one separating 4-cycle.

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