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Adjacency Graphs of Polyhedral Surfaces
Authors
+5Â more
Elena Arseneva
Linda Kleist
Boris Klemz
Birgit Vogtenhuber
Alexander Wolff
Publication date
1 January 2021
Publisher
LIPIcs - Leibniz International Proceedings in Informatics. 37th International Symposium on Computational Geometry (SoCG 2021)
Doi
Cite
View
on
arXiv
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
\mathbb{R}^3
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
K
5
â
,
K
5
,
81
K_{5,81}
K
5
,
81
â
, or any nonplanar
3
3
3
-tree as a subgraph, no such realization exists. On the other hand, all planar graphs,
K
4
,
4
K_{4,4}
K
4
,
4
â
, and
K
3
,
5
K_{3,5}
K
3
,
5
â
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
n
n
-vertex graphs is in
Ω
(
n
log
âĄ
n
)
\Omega(n \log n)
Ω
(
n
lo
g
n
)
. From the non-realizability of
K
5
,
81
K_{5,81}
K
5
,
81
â
, we obtain that any realizable
n
n
n
-vertex graph has
O
(
n
9
/
5
)
O(n^{9/5})
O
(
n
9/5
)
edges. As such, these graphs can be considerably denser than planar graphs, but not arbitrarily dense.Comment: To appear in Proc. SoCG 202
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Dagstuhl Research Online Publication Server
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oai:drops-oai.dagstuhl.de:1381...
Last time updated on 11/06/2021
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Last time updated on 20/03/2021