Star M6 and binary refinements. The special case of a refinement when the star M6 is present in a gene tree. An optimal edge can be found after rooting at the center node of star M6 and then applying the refinement procedure for rooted gene trees (see Algorithm 1). An optimal edge of every binary refinement of G is "surrounded" by the edges related to the star M6 present in G. For example, the candidates are two internal edges of G
for each i. The optimal binary refinement of G has the gene duplication cost equal to 0 and it is obtained by rooting G7 at the left internal edge. See also bottom part of this figure. Clearly, it has the same topology as the species tree. Similarly to Figure 3, each edge e of the gene tree G is decorated with the duplication cost D(G
, S), where G
is a (not refined) rooting of G.