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**Figure 1 A phylogenetic network**. The network was generated by Neighbor-Net for a sequence-based data set comprising of Salmonella isolates that originally appeared in [17]. A detailed network-based analysis of this data is presented in [2], where the strains indicated in bold-face are tested for the presence of recombination. Note that the network is planar (that is, it can be drawn in the plane without any crossing edges), and that parallel edges in the network represent bipartitions of the data.

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**Figure 2 The Neighbor-Net algorithm**. Pseudo-code for the Neighbor-Net algorithm detailing the procedure FINDORDERING.

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**Figure 3 A network representing a circular distance**. A circular distance d on the set {u, v, ..., z} for which NeighborNet using only the Q-criterion employed in NJ to cluster elements would be inconsistent. Distances are given by shortest paths in the network. The pairs u, v and x, y would be clustered together first and then the pair z, w. However it is not hard to show that z and w are not adjacent in any ordering of {u, v, ..., z} that is compatible with d.

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