Contact representations of planar graphs with cubes

Authors:
Stefan Felsner;Mathew C. Francis
Affiliations:
Technische Universität Berlin, Berlin, Germany;Charles University, Prague, Czech Rep
Venue:
Proceedings of the twenty-seventh annual symposium on Computational geometry
Year:
2011

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Abstract

We prove that every planar graph has a representation using axis-parallel cubes in three dimensions in such a way that there is a cube corresponding to each vertex of the planar graph and two cubes have a non-empty intersection if and only if their corresponding vertices are adjacent. Moreover, when two cubes have a non-empty intersection, they just touch each other. This result is a strengthening of a result by Thomassen which states that every planar graph has such a representation using axis-parallel boxes.