Geodesic
The number of labeled tetracyclic series-parallel~blocks
Prikladnaâ diskretnaâ matematika, no. 1 (2020), pp. 57-61

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A series-parallel graph is a graph that does not contain a complete graph with four vertices as a minor. Such graphs are used in the construction of reliable communication networks. Let $TB(n)$ be the number of labeled series-parallel tetracyclic blocks with $n$ vertices. The formula $TB(n)=\dfrac{n!}{80640}(n^5+30n^4+257n^3+768n^2+960n+504)\dbinom{n-3}{3}$ is obtained. It is proved that with a uniform probability distribution, the probability that the labeled tetracyclic block is a series-parallel graph is asymptotically $3/11$.
Keywords: labeled graph, tetracyclic graph, series-parallel graph, block, enumeration, asymptotics. 
@article{PDM_2020_1_a4,
     author = {V. A. Voblyi},
     title = {The number of labeled tetracyclic series-parallel~blocks},
     journal = {Prikladna\^a diskretna\^a matematika},
     pages = {57--61},
     publisher = {mathdoc},
     number = {1},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDM_2020_1_a4/}
}
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V. A. Voblyi. The number of labeled tetracyclic series-parallel~blocks. Prikladnaâ diskretnaâ matematika, no. 1 (2020), pp. 57-61. http://geodesic.mathdoc.fr/item/PDM_2020_1_a4/