Analytic combinatorics of non-crossing configurations

Flajolet, Philippe; Noy, Marc

doi:10.1016/s0012-365x(98)00372-0

Cited by 148 publications

(182 citation statements)

References 16 publications

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“…Hence, pg(S)/tr(S) = Θ * (2.9125 N ), whereas the upper bound provided by Theorem 3.1 is 3 N in this case. Informally, the (rather small) discrepancy between the exact bound in [11] and our bound in the convex case comes from the fact that when j is large, the faces of the resulting convex decomposition are likely to have many edges, which makes supp(G) substantially larger than 2 j . It is an interesting open problem to exploit this observation to improve our upper bound when h is large.…”

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

confidence: 80%

“…In this case we have tr(S) = Θ * (4 N ) and f(S) = Θ * (8.22 N ) (see [11]), so the exact ratio is f(S)/tr(S) = Θ * (2.055 N ), again suggesting that the ratio should be smaller when h is large.…”

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

confidence: 91%

“…In this case we have tr(S) = C N −2 = Θ * (4 N ), and pg(S) = Θ * (11.65 N ) (see [11]). Hence, pg(S)/tr(S) = Θ * (2.9125 N ), whereas the upper bound provided by Theorem 3.1 is 3 N in this case.…”

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

confidence: 97%

“…An extreme situation is when S is in convex position (in which case |F | = N − 3). In this case it is known that tr(S) = Θ * (4 N ) and st(S) = Θ * (6.75 N ) (see [11]), so the exact ratio is only st(S)/tr(S) = Θ * (1.6875 N ). This might suggest that when h is large the ratio should be considerably smaller, but we have not pursued this in this paper.…”

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

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

confidence: 80%

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

confidence: 91%

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

confidence: 97%

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

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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“…11.66 N ) [11]. The more general problem asks for the maximum number of crossing-free straight-edge graphs that can be embedded over any specific set of N points in the plane.…”

mentioning

confidence: 99%

Counting Plane Graphs: Cross-Graph Charging Schemes

Sharir

Sheffer

2013

Graph Drawing

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Abstract. We study cross-graph charging schemes for graphs drawn in the plane. These are charging schemes where charge is moved across vertices of different graphs. Such methods have been recently applied to obtain various properties of triangulations that are embedded over a fixed set of points in the plane. We show how this method can be generalized to obtain results for various other types of graphs that are embedded in the plane. Specifically, we obtain a new bound of 1 O * ( 187.53 N ) for the maximum number of crossing-free straight-edge graphs that can be embedded over any specific set of N points in the plane (improving upon the previous best upper bound 207.85 N in Hoffmann et al. [14]). We also derive upper bounds for numbers of several other types of plane graphs (such as connected and bi-connected plane graphs), and obtain various bounds on expected vertex-degrees in graphs that are uniformly chosen from the set of all crossing-free straight-edge graphs that can be embedded over a specific point set. We then show how to apply the cross-graph charging-scheme method for graphs that allow certain types of crossings. Specifically, we consider graphs with no set of k pairwise-crossing edges (more commonly known as k-quasi-planar graphs). For k = 3 and k = 4, we prove that, for any set S of N points in the plane, the number of graphs that have a straight-edge k-quasi-planar embedding over S is only exponential in N .

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