Representing Graphs and Hypergraphs by Touching Polygons in 3D

Evans, William S.; Rzążewski, Paweł; Saeedi, Noushin; Shin, Chan-Su; Wolff, Alexander

doi:10.1007/978-3-030-35802-0_2

Cited by 11 publications

(6 citation statements)

References 26 publications

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“…For proper side contacts, Kleist and Rahman [32] proved that every subgraph of an Archimedean grid can be represented with unit cubes, and every subgraph of a d-dimensional grid can be represented with d-cubes. Evans et al [19] showed that every graph has a contact representation where vertices are represented by convex polygons in R 3 and edges by shared corners of polygons, and gave polynomial-volume representations for bipartite, 1-planar, and cubic graphs.…”

Section: Contact Representationsmentioning

confidence: 99%

Adjacency Graphs of Polyhedral Surfaces

Arseneva,

Kleist,

Klemz

et al. 2023

Discrete Comput Geom

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We study whether a given graph can be realized as an adjacency graph of the polygonal cells of a polyhedral surface in $${\mathbb {R}}^3$$ 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 , $$K_{5,81}$$ K 5 , 81 , or any nonplanar 3-tree as a subgraph, no such realization exists. On the other hand, all planar graphs, $$K_{4,4}$$ K 4 , 4 , and $$K_{3,5}$$ 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 $$\Omega (n\log n)$$ Ω ( n log n ) . From the non-realizability of $$K_{5,81}$$ K 5 , 81 , we obtain that any realizable n-vertex graph has $${\mathcal {O}}(n^{9/5})$$ O ( n 9 / 5 ) edges. As such, these graphs can be considerably denser than planar graphs, but not arbitrarily dense.

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Section: Contact Representationsmentioning

confidence: 99%

Adjacency Graphs of Polyhedral Surfaces

Arseneva,

Kleist,

Klemz

et al. 2023

Discrete Comput Geom

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“…Gropp [32] positioned the vertices in the plane such that those that form hyperedges are collinear in the plane. Evans et al [33] used polygons to represent hyperedges in 3D to gain additional flexibility.…”

Section: Related Workmentioning

confidence: 99%

Topological Simplifications of Hypergraphs

Zhou¹,

Rathore²,

Purvine³

et al. 2021

Preprint

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We study hypergraph visualization via its topological simplification. We explore both vertex simplification and hyperedge simplification of hypergraphs using tools from topological data analysis. In particular, we transform a hypergraph to its graph representations known as the line graph and clique expansion. A topological simplification of such a graph representation induces a simplification of the hypergraph. In simplifying a hypergraph, we allow vertices to be combined if they belong to almost the same set of hyperedges, and hyperedges to be merged if they share almost the same set of vertices. Our proposed approaches are general, mathematically justifiable, and they put vertex simplification and hyperedge simplification in a unifying framework.

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“…For proper side contacts, Kleist and Rahman [25] proved that every subgraph of an Archimedean grid can be represented with unit cubes, and every subgraph of a d-dimensional grid can be represented with d-cubes. Evans et al [14] showed that every graph has a contact representation where vertices are represented by convex polygons in R 3 and edges by shared corners of polygons, and gave polynomial-volume representations for bipartite, 1-planar, and cubic graphs.…”

Section: Contact Representationsmentioning

confidence: 99%

“…Together with Proposition 2 this implies the claim. ◀ Evans et al [14] showed that every bipartite graph has a contact representation by touching polygons on a polynomial-size integer grid in R 3 for the case of corner contacts. As we have seen before, side contacts are less flexible.…”

Section: Complete Graphsmentioning

confidence: 99%

Adjacency Graphs of Polyhedral Surfaces

Arseneva¹,

Kleist²,

Klemz³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

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 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(n 9/5 ) edges. As such, these graphs can be considerably denser than planar graphs, but not arbitrarily dense. ACM Subject ClassificationMathematics of computing → Graphs and surfaces; Mathematics of computing → Combinatoric problems Keywords and phrases polyhedral complexes, realizability, contact representation Funding Elena Arseneva: partially supported by RFBR, project 20-01-00488. Boris Klemz: supported by DFG project WO 758/11-1. Birgit Vogtenhuber: partially supported by the Austrian Science Fund within the collaborative DACH project Arrangements and Drawings as FWF project I 3340-N35.

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Representing Graphs and Hypergraphs by Touching Polygons in 3D

Cited by 11 publications

References 26 publications

Adjacency Graphs of Polyhedral Surfaces

Adjacency Graphs of Polyhedral Surfaces

Topological Simplifications of Hypergraphs

Adjacency Graphs of Polyhedral Surfaces

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