Representing Graphs and Hypergraphs by Touching Polygons in 3D

  • William Evans
  • Paweł RzążewskiEmail author
  • Noushin Saeedi
  • Chan-Su Shin
  • Alexander Wolff
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11904)


Contact representations of graphs have a long history. Most research has focused on problems in 2d, but 3d contact representations have also been investigated, mostly concerning fully-dimensional geometric objects such as spheres or cubes. In this paper we study contact representations with convex polygons in 3d. We show that every graph admits such a representation. Since our representations use super-polynomial coordinates, we also construct representations on grids of polynomial size for specific graph classes (bipartite, subcubic). For hypergraphs, we represent their duals, that is, each vertex is represented by a point and each edge by a polygon. We show that even regular and quite small hypergraphs do not admit such representations. On the other hand, the two smallest Steiner triple systems can be represented.



We are grateful to the organizers of the workshop Homonolo 2017, where the project originates. We thank Günter Rote for advice regarding strictly convex drawings of polygons on the grid, and we thank Torsten Ueckerdt for bringing Ossona de Mendez’ work [21] to our attention. We are indebted to Arnaud de Mesmay and Eric Sedgwick for pointing us to the lemma of Dey and Edelsbrunner [9], which yielded Theorem 6.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.University of British ColumbiaVancouverCanada
  2. 2.Faculty of Mathematics and Information ScienceWarsaw University of TechnologyWarszawaPoland
  3. 3.Hankuk University of Foreign StudiesYonginRepublic of Korea
  4. 4.Universität WürzburgWürzburgGermany

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