A simple topological graph T = (V(T), E(T)) is a drawing of a graph in the plane where every two edges have at most one common point (an endpoint or a crossing) and no three edges pass through a single crossing. Topological graphs G and H are isomorphic if H can be obtained from G by a homeomorphism of the sphere, and weakly isomorphic if G and H have the same set of pairs of crossing edges. We generalize results of Pach and Toth and the author's previous results on counting different drawings of a graph under both notions of isomorphism. We prove that for every graph G with n vertices, m edges and no isolated vertices the number of weak isomorphism classes of simple topological graphs that realize G is at most 2^O(n^2 log(m/n)), and at most 2^O(mn^{1/2} log n) if m < n^{3/2}. As a consequence we obtain a new upper bound 2^O(n^{3/2} log n) on the number of intersection graphs of n pseudosegments. We improve the upper bound on the number of weak isomorphism classes of simple complete topological graphs with n vertices to 2^{n^2 alpha(n)^O(1)}, using an upper bound on the size of a set of permutations with bounded VC-dimension recently proved by Cibulka and the author. We show that the number of isomorphism classes of simple topological graphs that realize G is at most 2^{m^2+O(mn)} and at least 2^Omega(m^2) for graphs with m > (6+epsilon)n.
翻译:简单的表层图T = (V(T), E(T)) 是平面上一张图的图, 其中每两个边缘在最大一个共同点( 端点或交叉点) 下都有不同的图, 没有三个边缘通过一个单一交叉点。 地形图G 和 H 是 异形, 如果以球体的正态从 G 获得 H ; 如果 G 和 H 拥有相同的交叉边缘配对, 并且 弱异形 。 我们概括了 Pach 和 Toth 的结果, 以及作者先前在两个异形概念下计算图图的不同图画结果。 我们证明, 对于每个有正态、 m 边缘和无孤立的顶端的图形 G, 简单表层图的弱形类数量最多为 2°( n% 2) 2, 如果以 m < n% 1 = 2 和 c% 2 的正值, 我们最近获得一个新的上端的O% 2 (n) 3/2) 级的O 级图, 和正态的上层的正态, 的正态的正态将显示为n 。