On an approach to enumeration of labeled connected graphs: A review

The classical functional Riddell equation connects the generating functions of labeled connected graphs and their blocks. Using the Lagrange inversion theorem, the author obtained from this equation a formula, which is a convenient tool for the exact and asymptotic enumeration of labeled graphs in the case where the generating function of their blocks is known. This formula is valid for block-stable graph classes. A review of enumerative results obtained by using this approach for cacti, block-cactus graphs, Eulerian graphs, geodesic graphs, planar graphs, and series-parallel graphs is presented.