Fig. 3

(1) Left: a network G with 5 taxa at the leaves. The fat red edge is a transfer edge. Two taxa have no character, two have all characters of \(\mathcal {C}\), and one has only the odd numbered taxa (we assume odd d in the figure). Right: a \(\mathcal {C}\)-labeling that explains G (the reticulation receives the odd characters from the transfer edge). This network has no explanation with d-crossovers. (2) Left: a network G with 5 taxa on character set \(\mathcal {C}= \{c_1, c_2\}\). Right: a binary \(\mathcal {C}\)-labeling with single crossovers that explains G. This network is not a PTN