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Thirty Essays on Geometric Graph Theory pp 213–248Cite as

Rectangle and Square Representations of Planar Graphs

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Abstract

In the first part of this survey, we consider planar graphs that can be represented by a dissections of a rectangle into rectangles. In rectangular drawings, the corners of the rectangles represent the vertices. The graph obtained by taking the rectangles as vertices and contacts as edges is the rectangular dual. In visibility graphs and segment contact graphs, the vertices correspond to horizontal or to horizontal and vertical segments of the dissection. Special orientations of graphs turn out to be helpful when dealing with characterization and representation questions. Therefore, we look at orientations with prescribed degrees, bipolar orientations, separating decompositions, and transversal structures.

In the second part, we ask for representations by a dissections of a rectangle into squares. We review results by Brooks et al. [The dissection of rectangles into squares. Duke Math J 7:312–340 (1940)], Kenyon [Tilings and discrete Dirichlet problems. Isr J Math 105:61–84 (1998)], and Schramm [Square tilings with prescribed combinatorics. Isr J Math 84:97–118 (1993)] and discuss a technique of computing squarings via solutions of systems of linear equations.

Mathematics Subject Classification (2010): 05C10, 05C62, 52C15.

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Acknowledgements

I thank Thomas Picchetti for his implementation of the squaring algorithm of Sect. 5.1 and Julia Rucker and Torsten Ueckerdt for helpful discussions and their continuing interest in the topic. My thanks also go to Janos Pach for encouraging me to write about this topic and to the Bernoulli Centre for its hospitality.

This work was partially supported by DFG Grant FE-340/7-2 and the EUROGIGA Project GraDR.

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  1. Institut für Mathematik, Technische Universität Berlin, Berlin, Germany

    Stefan Felsner

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  1. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary

    János Pach (Chair of Combinatorial Geometry) (Chair of Combinatorial Geometry)

  2. École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    János Pach (Chair of Combinatorial Geometry) (Chair of Combinatorial Geometry)

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Felsner, S. (2013). Rectangle and Square Representations of Planar Graphs. In: Pach, J. (eds) Thirty Essays on Geometric Graph Theory. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0110-0_12

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