We derive precise asymptotic estimates for the number of labelled graphs not containing $K_{ 3,3}$ as a minor, and also for those which are edge maximal. Additionally, we establish limit laws for parameters in random $K_{ 3,3}$-minor-free graphs, like the expected number of edges. To establish these results, we translate a decomposition for the corresponding graph class into equations for generating functions and use singularity analysis. We also find a precise estimate for the number of graphs not containing the graph $K_{ 3,3}$ plus an edge as a minor.
Categories: math.CO