Enumeration of labeled series-parallel tricyclic graphs

A series-parallel graph is a graph that does not contain a complete graph with four vertices as a minor. An explicit formula for the number of labeled series-parallel tricyclic graphs with a given number of vertices is obtained, and the corresponding asymptotics for the number of such graphs with a large number of vertices is found. We prove that under a uniform probability distribution, the probability that the labeled tricyclic graph is a series-parallel graph is asymptotically equal to $13/15$.