Improved enumeration of simple topological graphs

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.

💡 Research Summary

The paper investigates the enumeration of simple topological graphs—drawings of a graph in the plane where any two edges intersect at most once and no three edges meet at a single crossing—under two notions of equivalence: topological isomorphism (homeomorphism of the sphere) and weak isomorphism (identical sets of crossing edge pairs). After recalling the definitions, the authors extend earlier results that were limited to complete graphs or to very coarse bounds for general graphs.

The first major theorem gives an upper bound on the number of weak‑isomorphism classes for any graph G with n vertices, m edges and no isolated vertices. By partitioning the edge set into O(m/n) groups and encoding the crossing pattern of each group as a permutation, they control the combinatorial explosion using recent bounds on the size of permutation families with bounded VC‑dimension. This yields a universal bound of 2^{O(n² log(m/n))}. When m < n^{3/2}, a refined partitioning argument improves the bound to 2^{O(m n^{1/2} log n)}. These results directly imply a new upper bound of 2^{O(n^{3/2} log n)} on the number of intersection graphs that can be realized by n pseudosegments, because such intersection graphs correspond bijectively to weak‑isomorphism classes of simple topological graphs.

The second contribution focuses on complete graphs K_n. Leveraging a recent theorem (Cibulka & the author) that limits the cardinality of a set of permutations whose VC‑dimension is k, the authors show that the VC‑dimension of the permutation families arising from K_n can be bounded by the inverse Ackermann function α(n). Consequently, the number of weak‑isomorphism classes of simple complete topological graphs is at most 2^{n² α(n)^{O(1)}}, a dramatic improvement over the previous 2^{O(n² log n)} bound. Since α(n) grows slower than any iterated logarithm, this bound is essentially quadratic in n up to a near‑constant factor.

For topological isomorphism, which is stricter because it records the exact cyclic order of edges around each vertex as well as the crossing order, the authors prove an upper bound of 2^{m²+O(mn)}. The argument observes that each unordered pair of edges can either cross or not, and if they cross there are only a constant number of possible relative orders, giving a factor of 2^{O(m²)}. The additional O(mn) term accounts for the choices of rotation systems at the vertices. On the lower‑bound side, they show that when the edge density exceeds (6+ε)n, one can construct graphs whose drawings realize 2^{Ω(m²)} distinct topological isomorphism classes, essentially matching the upper bound up to the lower‑order term.

The paper also discusses methodological innovations. The central technical tool is the application of VC‑dimension to permutation families derived from crossing patterns; this bridges combinatorial geometry and learning‑theoretic concepts. By controlling VC‑dimension, the authors limit the number of admissible permutations, which in turn caps the number of possible crossing configurations. The partitioning scheme for general graphs and the refined analysis for dense graphs (m > (6+ε)n) are tailored to exploit sparsity or density respectively.

In the concluding remarks, the authors note that the gap between the upper and lower bounds remains open, especially for the weak‑isomorphism case when m is close to n². They suggest that further tightening of VC‑dimension bounds or new structural decompositions could narrow this gap. The results have immediate implications for the study of intersection graphs of geometric objects, pseudosegment arrangements, and for algorithmic problems where the number of distinct topological embeddings influences complexity. Overall, the work significantly advances our understanding of how combinatorial parameters (n, m) control the richness of topological graph drawings.