Packing edge-disjoint cycles in graphs and the cyclomatic number

For a graph GG let μ(G)μ(G) denote the cyclomatic number and let ν(G)ν(G) denote the maximum number of edge-disjoint cycles of GG.We prove that for every k≥0k≥0 there is a finite set P(k)P(k) such that every 2-connected graph GG for which μ(G)−ν(G)=kμ(G)−ν(G)=k arises by applying a simple extension rule to a graph in P(k)P(k). Furthermore, we determine P(k)P(k) for k≤2k≤2 exactly.