Fig. 2

We show (a) \(T_1|_X\), (b) \(T_2|_X\), and (c) their incompatibility graph, based on the trees \(T_1\) and \(T_2\) in Fig. 1 (without the trivial bipartitions). Each \(\pi _i\) is the bipartition induced by \(e_i\), and the weights for \(\pi _1,\dots , \pi _8\) are 3, 4, 1, 1, 2, 2, 2, 3, in that order. We note that \(\pi _1\) and \(\pi _5\) are the same bipartition, but they have different weights as they are induced by different edges; similarly for \(\pi _3\) and \(\pi _7\). The maximum weight independent set in this graph has all the isolated vertices (\(\pi _1, \pi _3, \pi _5, \pi _7\)) and also \(\pi _2,\pi _8\), and so has total weight 15