Quasi-planar graphs have a linear number of edges

114

Section: Discussionmentioning

confidence: 69%

“…It is a well-known conjecture [3,Problem 3.3] that for any fixed k, there is a constant C k such that f k (n) ≤ C k n. Agarwal et al [2] were the first to prove this conjecture for k = 3. Later, Pach et al [4] simplified their proof and showed that f 3 (n) ≤ 65n.…”

Section: Introductionmentioning

confidence: 96%

On the Maximum Number of Edges in Topological Graphs with no Four Pairwise Crossing Edges

Ackerman

2009

114

“…There are many nice results for various forbidden classes-k pairwise crossing edges, k pairwise "parallel" edges, k pairwise disjoint edges, self-crossing paths, even cycles and many others (see [1], [2], [7], [8], [11], and [15]- [17]). For a survey of results on geometric graphs see [10].…”

Section: (H ) ⊆ V (G) and E(h ) ⊆ E(g)mentioning

confidence: 99%

Geometric Graphs with No Three Disjoint Edges

Černý

2005

“…In the second case, every edge e ∈ E that contains 2 as a face shares a vertex with 1 . Obviously, the number of such simplices e is at most ( 1 + 1) n d− 2 −1 .…”

Section: Full-dimensional Simplicesmentioning

confidence: 99%

Extremal Problems for Geometric Hypergraphs

Dey

Pach

1998

A geometric hypergraph H is a collection of i-dimensional simplices, called hyperedges or, simply, edges, induced by some (i + 1)-tuples of a vertex set V in general position in d-space. The topological structure of geometric graphs, i.e., the case d = 2, i = 1, has been studied extensively, and it proved to be instrumental for the solution of a wide range of problems in combinatorial and computational geometry. They include the k-set problem, proximity questions, bounding the number of incidences between points and lines, designing various efficient graph drawing algorithms, etc. In this paper, we make an attempt to generalize some of these tools to higher dimensions. We will mainly consider extremal problems of the following type. What is the largest number of edges (i-simplices) that a geometric hypergraph of n vertices can have without containing certain forbidden configurations? In particular, we discuss the special cases when the forbidden configurations are k intersecting edges, k pairwise intersecting edges, k crossing edges, k pairwise crossing edges, k edges that can be stabbed by an i-flat, etc. Some of our estimates are tight.