Adjacency graphs of polyhedral surfaces

Elena Arseneva*, Linda Kleist, Boris Klemz, Maarten Löffler, André Schulz, Birgit Vogtenhuber, Alexander Wolff

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference paperpeer-review

Abstract

We study whether a given graph can be realized as an adjacency graph of the polygonal cells of a polyhedral surface in ℝ3. We show that every graph is realizable as a polyhedral surface with arbitrary polygonal cells, and that this is not true if we require the cells to be convex. In particular, if the given graph contains K5, K5,81, or any nonplanar 3-tree as a subgraph, no such realization exists. On the other hand, all planar graphs, K4,4, and K3,5 can be realized with convex cells. The same holds for any subdivision of any graph where each edge is subdivided at least once, and, by a result from McMullen et al. (1983), for any hypercube. Our results have implications on the maximum density of graphs describing polyhedral surfaces with convex cells: The realizability of hypercubes shows that the maximum number of edges over all realizable n-vertex graphs is in Ω(n log n). From the non-realizability of K5,81, we obtain that any realizable n-vertex graph has O(n9/5) edges. As such, these graphs can be considerably denser than planar graphs, but not arbitrarily dense.

Original languageEnglish
Title of host publication37th International Symposium on Computational Geometry, SoCG 2021
EditorsKevin Buchin, Eric Colin de Verdiere
PublisherSchloss Dagstuhl - Leibniz-Zentrum für Informatik
ISBN (Electronic)9783959771849
DOIs
Publication statusPublished - 1 Jun 2021
Event37th International Symposium on Computational Geometry, SoCG 2021 - Virtual, Buffalo, United States
Duration: 7 Jun 202111 Jun 2021

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume189
ISSN (Print)1868-8969

Conference

Conference37th International Symposium on Computational Geometry, SoCG 2021
Country/TerritoryUnited States
CityVirtual, Buffalo
Period7/06/2111/06/21

Keywords

  • Contact representation
  • Polyhedral complexes
  • Realizability

ASJC Scopus subject areas

  • Software

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