Geodesic
On asymptotical enumeration of labeled series-parallel $k$-cyclic graphs
Diskretnyj analiz i issledovanie operacij, Tome 29 (2022) no. 4, pp. 5-14

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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.
Keywords: enumeration, labeled graph, block, series-parallel graph, $k$-cyclic graph, asymptotics, random graph. 
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     title = {On asymptotical enumeration of labeled series-parallel $k$-cyclic graphs},
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V. A. Voblyi. On asymptotical enumeration of labeled series-parallel $k$-cyclic graphs. Diskretnyj analiz i issledovanie operacij, Tome 29 (2022) no. 4, pp. 5-14. http://geodesic.mathdoc.fr/item/DA_2022_29_4_a0/