Homothetic triangle representations of planar graphs

Lévêque, Benjamin; Pinlou, Alexandre

doi:10.7155/jgaa.00509

“…If the prototypes and the curve are polygonal, i.e., are not smooth, then there still exists a contact representation, however, with the following shortcomings: The sets representing inner vertices may degenerate to points, which may lead to extra contacts. As observed by Gonçalves et al [24], Schramm's result implies that every subgraph of a 4-connected triangulation has a contact representation with aligned equilateral triangles and similarly, every inner triangulation of a 4-gon without separating 3-and 4-cycles has a hole-free contact representation with squares [20,43].…”

Section: Contact Representationsmentioning

confidence: 67%

Adjacency Graphs of Polyhedral Surfaces

Arseneva,

Kleist,

Klemz

et al. 2023

Discrete Comput Geom

0

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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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“…If the prototypes and the curve are polygonal, i.e., are not smooth, then there still exists a contact representation, however, the sets representing inner vertices may degenerate to points, which may lead to extra contacts. As observed by Gonçalves et al [19], Schramm's result implies that every subgraph of a 4-connected triangulation has a contact representation with aligned equilateral triangles and similarly, every inner triangulation of a 4-gon without separating 3and 4-cycles has a hole-free contact representation with squares [15].…”

Section: Contact Representationsmentioning

confidence: 67%

Adjacency Graphs of Polyhedral Surfaces

Arseneva¹,

Kleist²,

Klemz³

et al. 2021

Preprint

1

0

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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 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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“…Finally, Schienerman's conjecture was proved in 2009 by Chalopin & Gonc ¸alves [CG09]. More recently, Gonc ¸alves showed that planar graphs are intersection graphs of homothetic triangles [GLP19], and also of L-shapes [GIP18].…”

Section: Related Workmentioning

confidence: 98%

Finding Geometric Representations of Apex Graphs is NP-Hard

Chakraborty¹,

Gajjar²

2021

Preprint

0

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Planar graphs can be represented as intersection graphs of different types of geometric objects in the plane, e.g., circles (Koebe, 1936), line segments (Chalopin & Gonc ¸alves, 2009), L-shapes (Gonc ¸alves et al., 2018). Furthermore, these representations can be obtained in polynomial time when the planar graph is provided as input. For general graphs, however, even deciding whether such representations exist is often NP-hard. We consider apex graphs, i.e., graphs that can be made planar by removing one vertex from them. We show, somewhat surprisingly, that deciding whether geometric representations exist for apex graphs is NP-hard.More precisely, we show that recognizing every graph class G which satisfies PURE-2-DIR ⊆ G ⊆ 1-STRING is NP-hard, even when the input graphs are apex graphs. Here, PURE-2-DIR is the class of intersection graphs of axis-parallel line segments (where intersections are allowed only between horizontal and vertical segments) and 1-STRING is the class of intersection graphs of simple curves (where two curves share at most one point) in the plane. Most of the known NP-hardness reductions for these problems are from variants of 3-SAT. We reduce from PLANAR HAMILTONIAN PATH COMPLETION, which uses the more intuitive notion of planarity. As a result, our proof is much simpler and encapsulates several classes of geometric graphs.

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Homothetic triangle representations of planar graphs

Abstract: We prove that every planar graph is the intersection graph of homothetic triangles in the plane 1 .

Cited by 5 publications

References 19 publications

Adjacency Graphs of Polyhedral Surfaces

Adjacency Graphs of Polyhedral Surfaces

Adjacency Graphs of Polyhedral Surfaces

Finding Geometric Representations of Apex Graphs is NP-Hard

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