Packing edge-disjoint cycles in graphs and the cyclomatic number

Abstract

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