(**a**) A minimum rooted network *N* that represents the clusters = {{*a*}, {*b*}, {*c*}, {*d*}, {*o*}, {*h*}, {*a*, *b*}, {*a*, *b*, *h*}, {*b*, *h*}, {*c*, *d*}, {*c*, *d*, *h*}, {*a*, *b*, *c*, *d*, *h*}, {*a*, *b*, *c*, *d*}} using two reticulate nodes *r* and *s*. Note the role of the reticulate edge labeled *u* is to switch off the taxon *h* so that the cluster {*a*, *b*, *c*, *d*} can be represented by the tree edge *e*. (**b**) Two copies of *N* embedded into a decomposable network requiring four reticulations to represent all clusters in and a second set ′ of corresponding ones on {*o*′, *a*′, *b*′, *c*′, *d*′, *h*′}. (**c**) This network also represents all described clusters, but uses only three reticulations. However, we clearly see that this improvement is gained at an undesirable price: decomposability is abandoned (all clusters in are compatible with all clusters in ′, yet they are all represented in the same biconnected component of the network) and so two completely unrelated parts of the phylogeny are linked together via reticulation edges.

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