Abstract
Planar bipartite graphs can be represented as touching graphs of horizontal and vertical segments in \(\mathbb {R}^2\). We study a generalization in space, namely, touching graphs of axis-aligned rectangles in \(\mathbb {R}^3\). We prove that planar 3-colorable graphs can be represented as touching graphs of axis-aligned rectangles in \(\mathbb {R}^3\). The result implies a characterization of corner polytopes previously obtained by Eppstein and Mumford. A by-product of our proof is a distributive lattice structure on the set of orthogonal surfaces with given skeleton.
Moreover, we study the subclass of strong representations, i.e., families of axis-aligned rectangles in \(\mathbb {R}^3\) in general position such that all regions bounded by the rectangles are boxes. We show that the resulting graphs correspond to octahedrations of an octahedron. This generalizes the correspondence between planar quadrangulations and families of horizontal and vertical segments in \(\mathbb {R}^2\) with the property that all regions are rectangles.
Omitted proofs and more figures can be found in the full version [11].
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Notes
- 1.
We use the term touching graphs rather than the more standard contact graph to underline the fact that segments with coinciding endpoints (e.g., two horizontal segments touching a vertical segment in the same point but from different sides, but also non-parallel segments with coinciding endpoint) do not form an edge.
- 2.
Plattenbau (plural Plattenbauten) is a German word describing a building (Bau) made of prefabricated concrete panels (Platte).
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Felsner, S., Knauer, K., Ueckerdt, T. (2020). Plattenbauten: Touching Rectangles in Space. In: Adler, I., Müller, H. (eds) Graph-Theoretic Concepts in Computer Science. WG 2020. Lecture Notes in Computer Science(), vol 12301. Springer, Cham. https://doi.org/10.1007/978-3-030-60440-0_13
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