The number of graphs not containing K3,3 as a minor

We derive precise asymptotic estimates for the number of labelled graphs not containing K-3,K-3 as a minor, and also for those which are edge maximal. Additionally, we establish limit laws for parameters in random K-3,K-3-minor-freegraphs, like the number of edges. To establish these results, we translate a decomposition for the corresponding graphs 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,K-3 plus an edge as a minor