Adjacency Graphs of Polyhedral Surfaces.
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| Title: | Adjacency Graphs of Polyhedral Surfaces. |
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| 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] |
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| Database: | Engineering Source |
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| 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] |
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| ISSN: | 01795376 |
| DOI: | 10.1007/s00454-023-00537-6 |
