Bounds on the Maximum Multiplicity of Some Common Geometric Graphs

Dumitrescu, Adrian; Schulz, André; Sheffer, Adam; Tóth, Csaba D.

doi:10.1137/110849407

Cited by 42 publications

(47 citation statements)

References 26 publications

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“…We consider important to note that configurations of points having from Ω (3.464 n ), see [24], to Ω (8.65 n ), see [25], triangulations are known. Thus the algorithm presented in this paper counts triangulations faster than enumeration algorithms in at least those cases.…”

Section: Discussionmentioning

confidence: 99%

Counting Triangulations and Other Crossing-Free Structures via Onion Layers

Álvarez

Bringmann

Curticapean

et al. 2015

Discrete Comput Geom

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Let P be a set of n points in the plane. A crossing-free structure on P is a plane graph with vertex set P . Examples of crossing-free structures include triangulations of P , spanning cycles of P , also known as polygonalizations of P , among others. There has been a large amount of research trying to bound the number of such structures. In particular, bounding the number of (straight-edge) triangulations spanned by P has received considerable attention. It is currently known that every set of n points has at most O(30 n ) and at least Ω(2.43 n ) triangulations. However, much less is known about the algorithmic problem of counting crossing-free structures of a given set P . For example, no algorithm for counting triangulations is known that, on all instances, performs faster than enumerating all triangulations.In this paper we develop a general technique for computing the number of crossing-free structures of an input set P . We apply the technique to obtain algorithms for computing the number of triangulations, matchings, and spanning cycles of P . The running time of our algorithms is upper bounded by n O(k) , where k is the number of onion layers of P . In particular, for k = O(1) our algorithms run in polynomial time. In addition, we show that our algorithm for counting triangulations is never slower than O * (3.1414 n ), even when k = Θ(n). Given that there are several well-studied configurations of points with at least Ω(3.464 n ) triangulations, and some even with Ω(8 n ) triangulations, our algorithm asymptotically outperforms any enumeration algorithm for such instances, and it has better worst-case behavior than the recent algorithm shown in [1], which also beats enumeration in those instances. In fact, it is widely believed that any set of n points must have at least Ω(3.464 n ) triangulations. If this is true, then our algorithm is strictly sub-linear in the number of triangulations counted. We also show that our techniques are general enough to solve the Restricted-Triangulation-Counting-Problem, which we prove to be W [2]-hard in the parameter k. This implies a "no free lunch" result: In order to be fixed-parameter tractable, our general algorithm must rely on additional properties that are specific to the considered class of structures.

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Section: Discussionmentioning

confidence: 99%

Counting Triangulations and Other Crossing-Free Structures via Onion Layers

Álvarez

Bringmann

Curticapean

et al. 2015

Discrete Comput Geom

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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%

“…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%

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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“…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]). Worst-case lower bounds for these numbers have also been addressed (e.g., see [3,9,12]). 2 Research on the above problems has led to the development of several useful combinatorial techniques, many of which are interesting in their own right.…”

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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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Bounds on the Maximum Multiplicity of Some Common Geometric Graphs

Cited by 42 publications

References 26 publications

Counting Triangulations and Other Crossing-Free Structures via Onion Layers

Counting Triangulations and Other Crossing-Free Structures via Onion Layers

Convex Polygons in Geometric Triangulations

Counting Plane Graphs: Cross-Graph Charging Schemes

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