On asymptotical enumeration of labeled series-parallel $k$-cyclic graphs

We deduce an asymptotic formula for the number of labeled connected series-parallel $k$-cyclic graphs with given order and fixed number $k$. Under uniform probability distribution, we find the probability that a random labeled connected $n$-vertex $k$-cyclic graph with a fixed $k$ and $n\to\infty$ is a series-parallel graph. In addition, we determine the probability that, under uniform probability distribution, a random labeled connected series-parallel $n$-vertex $k$-cyclic graph with a fixed $k$ and $n\to\infty$ is a cactus. Bibliogr. 16.