Number of Crossing-Free Geometric Graphs vs. Triangulations

Razen, Andreas; Snoeyink, Jack; Welzl, Emo

doi:10.1016/j.endm.2008.06.039

Cited by 15 publications

(20 citation statements)

References 5 publications

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“…We also let tr(N ) and pg(N ) denote, respectively, the maximum values of tr(S) and of pg(S), over all sets S of N points in the plane. Since a triangulation of S has fewer than 3|S| edges, the trivial upper bound pg(S) < 8 |S| · tr(S) holds for any point set S. Recently, Razen, Snoeyink, and Welzl [22] slightly improved the upper bound on the ratio pg(S)/tr(S) from 8 |S| down to O 7.9792 |S| . We give a more significant improvement on the ratio with an upper bound of 6.9283 |S| .…”

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: Introductionmentioning

confidence: 99%

“…Thus, every graph G ∈ P(S) will be counted supp(G) times in the preceding inequality. Recently, Razen, Snoeyink, and Welzl [22] managed to break the 8 N barrier by overcoming the above inefficiency. However, they obtained only a slight improvement, with the bound pg(S) = O 7.9792 N · tr(S).…”

Section: Pg(s) < 8 N · Tr(s)mentioning

confidence: 99%

Counting Plane Graphs: Flippability and Its Applications

Hoffmann

Sharir

Sheffer

et al. 2011

Lecture Notes in Computer Science

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“…The number of crossing-free structures (matchings, spanning trees, spanning cycles, triangulations) on a set of n points in the plane is known to be exponential in n [11,16,21,24,25,26]. It is a challenging problem to determine the number of configurations faster than listing all such configurations (i.e., count faster than enumerate) [3].…”

Section: Counting Algorithmmentioning

confidence: 99%

“…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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“…The first exponential bound, 10 13N , on the number of such graphs was proved by Ajtai et al [4] back in 1982. Since then, progressively (and significantly) smaller upper bounds have been derived (for example, see [14,18,23]). Upper bounds on numbers of more specific types of crossing-free straight-edge graphs, such as Hamiltonian cycles, spanning trees, perfect matchings, and triangulations, were also studied (e.g., see [6,7,20,21,25]).…”

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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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Number of Crossing-Free Geometric Graphs vs. Triangulations

Cited by 15 publications

References 5 publications

Counting Plane Graphs: Flippability and Its Applications

Counting Plane Graphs: Flippability and Its Applications

Convex Polygons in Geometric Triangulations

Counting Plane Graphs: Cross-Graph Charging Schemes

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