计算平面图：交叉图计费方案
Counting Plane Graphs: Cross-Graph Charging Schemes
摘要：
我们研究了平面图的交图充电方案。这些是电荷在不同图的顶点之间移动的充电方案。这种方法最近已被应用于获得各种属性的三角剖分是嵌入在一组固定的点在平面上。我们展示了如何推广这种方法，以获得各种其他类型的图形嵌入在平面上的结果。具体来说，我们得到了一个新的界的$O^*(187.53^N)$（其中$O^*()$符号隐藏多项式因子）的最大数量的交叉自由直边图，可以嵌入在平面上的任何特定的一组$N$点（改进后，以前的最佳上限$207.85^N$在霍夫曼等人）。我们还得到了其他几种类型的平面图（如连通和双连通平面图）的数量的上界，并获得了各种边界的预期顶点度的图，均匀地从一组所有交叉自由直边图，可以嵌入到一个特定的点集。然后，我们将展示如何应用交叉图收费计划的方法，允许某些类型的交叉点的图形。具体来说，我们考虑没有一组$k$成对交叉边的图（更常见的称为$k$ -准平面图）。对于$k=3$和$k=4$，我们证明了：对于平面上$N$点的任意集合$S$，在$S$上具有直边$k$ -拟平面嵌入的图的个数在$N$中仅是指数的.
英文摘要：
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 $O^*(187.53^N)$ (where the $O^*()$ notation hides polynomial factors) 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.). 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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