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The second Riddell relation relates the generating functions for the number of labeled connected graphs and the number of labeled blocks. We consider the conditions under which this relation is true for a subclass of connected graphs.Under these conditions, the formulas are valid that express the number of graphs from a subclass of labeled connected graphs trough the generating function of their blocks. By way of application, we obtain expressions for the numbers of labeled connected and 2-connected series-parallel graphs. [ABSTRACT FROM AUTHOR]

A k-cyclic graph is a graph with cyclomatic number k. An explicit formula for the number of labeled connected outerplanar k-cyclic graphs with a given number of vertices is obtained. In addition, such graphs with fixed cyclomatic number k and a large number of vertices are asymptotically enumerated. As a consequence, it is found that, for fixed k, almost all labeled connected outerplanar k-cyclic graphs with a large number of vertices are cacti. [ABSTRACT FROM AUTHOR]

The article discusses on how to obtain the closed-form expressions for the number of labeled bi- and tricyclic graphs and the asymptotics of the number of such graphs. It says that the cyclomatic number of any of those graphs is k +1. It mentions that E. F. Dmitriev used other methods in enumerating abeled 2-edge-connected graphs but has not published his results.