Invariants of Graph Drawings in the Plane

Skopenkov, A.

doi:10.1007/s40598-019-00128-5

Cited by 12 publications

(5 citation statements)

References 49 publications

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“…Investigating the computational complexity for the remaining open entries in Table 2 remains for future work. We strengthen the conjecture of Skopenkov [54] as follows.…”

Section: Discussionsupporting

confidence: 55%

Geometric Embeddability of Complexes is $\exists \mathbb R$-complete

Abrahamsen¹,

Kleist²,

Miltzow³

2021

Preprint

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We show that the decision problem of determining whether a given (abstract simplicial) k-complex has a geometric embedding in R d is complete for the Existential Theory of the Reals for all d ≥ 3 and k ∈ {d − 1, d}. This implies that the problem is polynomial time equivalent to determining whether a polynomial equation system has a real root. Moreover, this implies NP-hardness and constitutes the first hardness results for the algorithmic problem of geometric embedding (abstract simplicial) complexes.

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“…Investigating the computational complexity for the remaining open entries in Table 2 remains for future work. We strengthen the conjecture of Skopenkov [54] as follows.…”

Section: Discussionsupporting

confidence: 55%

Geometric Embeddability of Complexes is $\exists \mathbb R$-complete

Abrahamsen¹,

Kleist²,

Miltzow³

2021

Preprint

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“…The algorithmic problem of determining whether a given kdimensional (abstract) simplicial complex embeds in R d is an active field of research [11,22,37,39,45,46]. There exist at least three interesting notions of embeddability: linear, piecewise linear, and topological embeddability, which usually are not the same [37].…”

Section: Simplicial Complexesmentioning

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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“…It is known that for G = K 5 or G = K 3,3 , the number of all disjoint crossings of a plane generic immersion of G is always odd. See for example [10,Proposition 2.1] or [13,Lemma 1.4.3]. Some theorems on plane immersed graphs are also stated in [13].…”

Section: Introductionmentioning

confidence: 99%

“…See for example [10,Proposition 2.1] or [13,Lemma 1.4.3]. Some theorems on plane immersed graphs are also stated in [13]. See also [2] for related results.…”

Section: Introductionmentioning

confidence: 99%

Crossing numbers and rotation numbers of cycles in a plane immersed graph

Inoue,

Kimura,

Nikkuni

et al. 2022

Preprint

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For any generic immersion of a Petersen graph into a plane, the number of crossing points between two edges of distance one is odd. The sum of the crossing numbers of all 5-cycles is odd. The sum of the rotation numbers of all 5-cycles is even. We show analogous results for 6-cycles, 8-cycles and 9cycles. For any Legendrian spatial embedding of a Petersen graph, there exists a 5-cycle that is not an unknot with maximal Thurston-Bennequin number, and the sum of all Thurston-Bennequin numbers of the cycles is 7 times the sum of all Thurston-Bennequin numbers of the 5-cycles. We show analogous results for a Heawood graph. We also show some other results for some graphs. We characterize abstract graphs that has a generic immersion into a plane whose all cycles have rotation number 0.

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Invariants of Graph Drawings in the Plane

Cited by 12 publications

References 49 publications

Geometric Embeddability of Complexes is $\exists \mathbb R$-complete

Geometric Embeddability of Complexes is $\exists \mathbb R$-complete

Adjacency Graphs of Polyhedral Surfaces

Crossing numbers and rotation numbers of cycles in a plane immersed graph

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