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Adjacency Graphs of Polyhedral Surfaces.

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Title: Adjacency Graphs of Polyhedral Surfaces.
Authors: Arseneva, Elena1 (AUTHOR), Kleist, Linda2 (AUTHOR), Klemz, Boris3 (AUTHOR) Boris.Klemz@uni-wuerzburg.de, Löffler, Maarten4 (AUTHOR), Schulz, André5 (AUTHOR), Vogtenhuber, Birgit6 (AUTHOR), Wolff, Alexander3 (AUTHOR)
Source: Discrete & Computational Geometry. Jun2024, Vol. 71 Issue 4, p1429-1455. 27p.
Subjects: Convex surfaces, Polyhedral functions, Hypercubes, Planar graphs
Geographic Terms: Israel
Abstract: We study whether a given graph can be realized as an adjacency graph of the polygonal cells of a polyhedral surface in R 3 . 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 K 5 , K 5 , 81 , or any nonplanar 3-tree as a subgraph, no such realization exists. On the other hand, all planar graphs, K 4 , 4 , and K 3 , 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. (Isr. J. Math. 46(1–2), 127–144 (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 K 5 , 81 , we obtain that any realizable n-vertex graph has O (n 9 / 5) edges. As such, these graphs can be considerably denser than planar graphs, but not arbitrarily dense. [ABSTRACT FROM AUTHOR]
Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: We study whether a given graph can be realized as an adjacency graph of the polygonal cells of a polyhedral surface in R 3 . 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 K 5 , K 5 , 81 , or any nonplanar 3-tree as a subgraph, no such realization exists. On the other hand, all planar graphs, K 4 , 4 , and K 3 , 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. (Isr. J. Math. 46(1–2), 127–144 (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 K 5 , 81 , we obtain that any realizable n-vertex graph has O (n 9 / 5) edges. As such, these graphs can be considerably denser than planar graphs, but not arbitrarily dense. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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      – SubjectFull: Planar graphs
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              Text: Jun2024
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