Enumeration of labeled nonplanar pentacyclic blocks

An planar graph is a graph that can be drawn on a plane without intersecting edges. A pentacyclic graph is a connected graph with $n$ vertices and $n + 4$ edges. We obtain an explicit formula for the number of labeled nonplanar pentacyclic blocks with a given number of vertices and found the corresponding asymptotics for the number of such graphs with a large number of vertices. We prove that under the uniform probability distribution, the probability that the labeled pentacyclic block is a nonplanar graph is asymptotically equal to $80/539$.