On the Number of Plane Geometric Graphs

Aichholzer, Oswin; Hackl, Thomas; Huemer, Clemens; Hurtado, Ferrán; Krasser, Hannes; Vogtenhuber, Birgit

doi:10.1007/s00373-007-0704-5

Cited by 58 publications

(101 citation statements)

References 6 publications

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“…Moreover, we let f(N ) = max |S|=N f(S). Buchin and Schulz [6] have recently shown that every crossingfree straight-edge graph contains O 6.4884 N forests (improving a simple upper bound of O * (6.75 N ) observed in [1]). Following the approach of [6], we combine the bounds for spanning trees (just established) and for plane graphs with a bounded number of edges (established in Section 3.3 below), to obtain the following result.…”

Section: Proof the Exact Value Of St(s) Ismentioning

confidence: 76%

“…For a lower bound on pg(S)/tr(S), we consider the double chain configurations, presented in [14] (and depicted in Figure 8). It is shown in [14] that, when S is a double chain configuration, tr(S) = Θ * 8 N and pg(S) = Θ * 39.8 N (actually, only the lower bound on pg(N ) is given in [14]; the upper bound appears in [1]). Thus, we have pg(S) = Θ * 4.975 N · tr(S) (for this set h = 4, so h has no real effect on the asymptotic bound of Theorem 3.1).…”

Section: Theorem 31 For Every Set S Of N Points In the Plane H Of Wmentioning

confidence: 99%

“…To bound f(S), we evaluate max k min{g 1 (N, k), g 2 (N, k)}. A numerical calculation shows that the maximum value is obtained for k ′ ≈ 0.0285N , and the theorem follows since min{g 1 …”

Section: Proof the Exact Value Of St(s) Ismentioning

confidence: 99%

“…The initial upper bound 10 13N of [2] has been steadily improved in several paper (e.g., see [8,25,29]), culminating with the current record of 30 N due to Sharir and Sheffer [26]. Other papers have studied lower bounds on the maximal number of triangulations (e.g., [1,9]), and upper or lower bounds on the number of other kinds of planar graphs (e.g., [5,6,23,24]). …”

Section: Introductionmentioning

confidence: 99%

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Counting Plane Graphs: Flippability and Its Applications

Hoffmann

Sharir

Sheffer

et al. 2011

Lecture Notes in Computer Science

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We generalize the notions of flippable and simultaneously flippable edges in a triangulation of a set S of points in the plane to so-called pseudo-simultaneously flippable edges. Such edges are related to the notion of convex decompositions spanned by S.We prove a worst-case tight lower bound for the number of pseudo-simultaneously flippable edges in a triangulation in terms of the number of vertices. We use this bound for deriving new upper bounds for the maximal number of crossing-free straight-edge graphs that can be embedded on any fixed set of N points in the plane. We obtain new upper bounds for the number of spanning trees and forests as well. Specifically, let tr(N ) denote the maximum number of triangulations on a set of N points in the plane. Then we show (using the known bound tr(N ) < 30 N ) that any N -element point set admits at most 6. forests. We also obtain upper bounds for the number of crossing-free straight-edge graphs that have cN , fewer than cN , or more than cN edges, for any constant parameter c, in terms of c and N .

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Section: Proof the Exact Value Of St(s) Ismentioning

confidence: 76%

Section: Theorem 31 For Every Set S Of N Points In the Plane H Of Wmentioning

confidence: 99%

Section: Proof the Exact Value Of St(s) Ismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Counting Plane Graphs: Flippability and Its Applications

Hoffmann

Sharir

Sheffer

et al. 2011

Lecture Notes in Computer Science

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“…Analogous problems have been previously studied for cycles, spanning cycles, spanning trees, and matchings [6] in n-vertex edge-maximal planar graphs-that are defined in purely graph theoretic terms. For plane straight-line graphs, previous research focused on the maximum number of (noncrossing) configurations such as plane graphs, spanning trees, spanning cycles, triangulations, and others, over all n-element point sets in the plane [1,2,11,16,21,23,24,25,26]; see also the two surveys [12,27]. Early upper bounds in this area were obtained by multiplying the maximum number of triangulations on n point in the plane with the maximum number of desired configurations in an n-vertex triangulation, based on the fact that every planar straight-line graph can be augmented into a triangulation.…”

Section: Theoremmentioning

confidence: 99%

Convex Polygons in Geometric Triangulations

Dumitrescu

Tóth

2015

Lecture Notes in Computer Science

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We show that the maximum number of convex polygons in a triangulation of n points in the plane is O(1.5029 n ). This improves an earlier bound of O(1.6181 n ) established by van Kreveld, Löffler, and Pach (2012) and almost matches the current best lower bound of Ω(1.5028 n ) due to the same authors. Given a planar straight-line graph G with n vertices, we also show how to compute efficiently the number of convex polygons in G.

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Counting Plane Graphs: Flippability and Its Applications

Hoffmann

Sharir

Sheffer

et al. 2012

Thirty Essays on Geometric Graph Theory

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We generalize the notions of flippable and simultaneously flippable edges in a tri-angulation of a set S of points in the plane to so-called pseudo-simultaneously flippable edges. Such edges are related to the notion of convex decompositions spanned by S. We prove a worst-case tight lower bound for the number of pseudo-simultaneously flippable edges in a triangulation in terms of the number of vertices. We use this bound for deriving new upper bounds for the maximal number of crossing-free straight-edge graphs that can be embedded on any fixed set of N points in the plane. We obtain new upper bounds for the number of spanning trees and forests as well. Specifically, let tr(N) denote the maximum number of triangulations on a set of N points in the plane. Then we show (using the known bound tr(N) < 30 N) that any N-element point set admits at most 6.9283 N · tr(N) < 207.85 N crossing-free straight-edge graphs, O(4.7022 N) · tr(N) = O(141.07 N) spanning trees, and O(5.3514 N) · tr(N) = O(160.55 N) forests. We also obtain upper bounds for the number of crossing-free straight-edge graphs that have cN , fewer than cN , or more than cN edges, for any constant parameter c, in terms of c and N .

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On the Number of Plane Geometric Graphs

Cited by 58 publications

References 6 publications

Counting Plane Graphs: Flippability and Its Applications

Counting Plane Graphs: Flippability and Its Applications

Convex Polygons in Geometric Triangulations

Counting Plane Graphs: Flippability and Its Applications

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