Asymptotical enumeration of labeled series-parallel tetracyclic graphs

A series-parallel graph is a graph that does not contain a complete graph with four vertices as a minor. We find an asymptotics for the number of labeled connected series-parallel tetracyclic graphs with a large number of vertices. We prove that under a uniform probability distribution, the probability of the fact that a labeled connected tetracyclic graph is a series-parallel graph is asymptotically equal to $141/221$.