TY - GEN
T1 - Adjacency graphs of polyhedral surfaces
AU - Arseneva, Elena
AU - Kleist, Linda
AU - Klemz, Boris
AU - Löffler, Maarten
AU - Schulz, André
AU - Vogtenhuber, Birgit
AU - Wolff, Alexander
N1 - Funding Information:
Funding Elena Arseneva: partially supported by RFBR, project 20-01-00488. Boris Klemz: supported by DFG project WO758/11-1. Birgit Vogtenhuber: partially supported by the Austrian Science Fund within the DACH project Arrangements and Drawings as FWF project I 3340-N35.
Publisher Copyright:
© Elena Arseneva, Linda Kleist, Boris Klemz, Maarten Löffler, André Schulz, Birgit Vogtenhuber, and Alexander Wolff; licensed under Creative Commons License CC-BY 4.0 37th International Symposium on Computational Geometry (SoCG 2021).
PY - 2021/6/1
Y1 - 2021/6/1
N2 - 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.
AB - 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.
KW - Contact representation
KW - Polyhedral complexes
KW - Realizability
UR - http://www.scopus.com/inward/record.url?scp=85108234889&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.SoCG.2021.11
DO - 10.4230/LIPIcs.SoCG.2021.11
M3 - Conference paper
AN - SCOPUS:85108234889
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 37th International Symposium on Computational Geometry, SoCG 2021
A2 - Buchin, Kevin
A2 - de Verdiere, Eric Colin
PB - Schloss Dagstuhl - Leibniz-Zentrum für Informatik
T2 - 37th International Symposium on Computational Geometry, SoCG 2021
Y2 - 7 June 2021 through 11 June 2021
ER -