Counting Plane Graphs: Cross-Graph Charging Schemes

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 .

A Census of Plane Graphs with Polyline Edges

“…Given a plane graph G = (S, E) with k bends per edge, we replace each edge with a homotopic shortest path. The union of the shortest paths forms a plane straight-line graph G = (S, E ) on S. Note that the shortest paths may overlap, and G may be obtained in this way from several k-bend plane graphs on S. Theorem 1 follows from an 2 O(n) bound on the number of plane straight-line graphs G = (S, E ) [23], combined with an 2 O(kn) bound on the number of k-bend plane graphs that correspond to a straight-line graph G = (S, E ). The latter bound is obtained by encoding the shortest paths homotopic to the kbend edges of G with O(kn) bits of information.…”

“…13 . This upper bound has been improved successively over the last decades: the current best upper bound O(187.53 n ) is due to Sharir and Sheffer [23], using a so-called cross-graph charging scheme [21,24]. The current best lower bound, Ω(41.18 n ), is due to Aichholzer et al [1].…”

“…Given a plane graph G = (S, E) with k-bend edges, we could introduce new vertices at each bend point, and obtain a plane straight-line graph on at most n + (3n − 6)k = O(kn) vertices. We know that any set of m = O(kn) points admits O(187.53 m ) = 2 O(kn) plane straight-line graphs [23]. This bound, however, assumes that the set of bend points is fixed.…”

Monotone Paths in Geometric Triangulations

“…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 [2,7,10,19,22,23,24,25]; see also [8,26]. Early upper bounds in this area were obtained by multiplying an upper bound on the maximum number of triangulations on n points with an upper bound on the maximum number of desired configurations in an n-vertex triangulation; valid upper bounds result since every plane geometric graph can be augmented into a triangulation.…”

A simple aggregative algorithm for counting triangulations of planar point sets and related problems

“…For all these classes exponential lower and upper bounds have been proven for their sizes [4,24,13,3,18,27,26,14,25,23,2], i.e. statements of the form "for any planar set S of n points the number of plane perfect matchings is Ω(c n 1 ) and O(c n 2 )."…”