The Number of Labeled Outerplanar $k$-Cyclic Graphs

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.