Asymptotical enumeration of some abeled geodetic graphs

We asymptotically enumerate labeled geodetic $k$-cyclic cacti and obtain asymptotics for the numbers of labeled connected geodetic unicyclic, bicyclic, and tricyclic $n$-vertex graphs. We prove that under the uniform probability distribution, the probabilities that a random labeled connected unicyclic, bicyclic, or tricyclic graph is a geodetic graph are asymptotically equal to $1/2$, $3/20$, and $1/30$, respectively. In addition, we prove that almost all labeled connected geodetic tricyclic graphs are cacti.