Fig. 1.

A phylogenetic network on taxa within the $\gamma$-proteobacteria class with two reticulate events (Nakhleh et al. 2005). The dotted arcs with the arrowheads illustrate the passing of genetic material via a horizontal transfer. In this and all subsequent figures, the non-horizontal arcs without arrowheads are directed downwards. The network is a binary orchard network, since it can be reduced by a sequence of cherry reductions (see Sect. 2.1 for formal definitions of such reductions). One possible sequence that would reduce the network is  (1, 2)(3, 4)(2, 4)(5, 6)(4, 6)(7, 8)(9, 8)(10, 6)(6, 8)(8, 9)(9, 10)(11, 12)(12, 13)(10, 13). An HGT-consistent labelling of the network is given, indicated by the number next to each vertex. Note that the endpoints of an arc have the same label precisely if the arc corresponds to a horizontal transfer