Finally, if G is biconnected, then we make use of the open ear decomposition on the subgraph consisting of the heavy vertices only. If there is an ear with at least one interior (heavy) vertex, it suffices to construct the corresponding path by recombination of the growing chain with the star graph in a single step. Once the path is available it is recombined into G. If there are only ears left that consist of a single edge (i.e., that do not have interior vertices) between to heavy vertices, uv, then we consider the subgraph G ′ from which uv is cut out (and loose ends are compensated by degree one vertices, and the fragment uv). Cutting the requisite number of degree 1 vertices from G ′ and uv, respectively, there is a degree-preserving join that produced G. The insertion of an edge thus can be achieved with two cut-and-join steps: one for the formation of uv and one for the insertion of uv.