Lemma 4. Suppose removal of the cut D ⊆ E(G) decomposes G into three or more connected components. Then D is redundant. Suppose we restrict ourselves to 2-cuts and sets of graphs without cycles, i.e. trees. Then every cut-set contains only a single edge and thus all possible degree-preserving joins are again trees. Similarly, if all graphs are trees or cacti, i.e., graphs in which every cycle forms its own block (2-connected component), then all 2-cuts either contain individual edges or cut a unique cycle in exactly 2 points. Any edge-preserving join therefore is again a cactus. A restriction to 2-cuts thus severely limits the generative power of degreepreserving crossover operators. We will have a close look at this issue below.