Analytic combinatorics of non-crossing configurations
Discrete Mathematics - Special issue on selected papers in honor of Henry W. Gould
Lower bounds on the number of crossing-free subgraphs of KN
Computational Geometry: Theory and Applications
Graph Drawings with no k Pairwise Crossing Edges
GD '97 Proceedings of the 5th International Symposium on Graph Drawing
A better upper bound on the number of triangulations of a planar point set
Journal of Combinatorial Theory Series A
Random triangulations of planar point sets
Proceedings of the twenty-second annual symposium on Computational geometry
Note: On the maximum number of edges in quasi-planar graphs
Journal of Combinatorial Theory Series A
On the Number of Plane Geometric Graphs
Graphs and Combinatorics
On the Maximum Number of Edges in Topological Graphs with no Four Pairwise Crossing Edges
Discrete & Computational Geometry
On the number of spanning trees a planar graph can have
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
On degrees in random triangulations of point sets
Journal of Combinatorial Theory Series A
Counting plane graphs with exponential speed-up
Rainbow of computer science
Counting plane graphs: flippability and its applications
WADS'11 Proceedings of the 12th international conference on Algorithms and data structures
On the number of cycles in planar graphs
COCOON'07 Proceedings of the 13th annual international conference on Computing and Combinatorics
Computational geometry column 54
ACM SIGACT News
On numbers of pseudo-triangulations
Computational Geometry: Theory and Applications
| Hi-index | 0.00 |
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 $O^*\left(187.53^N \right)$ 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 207.85N 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.