The Dominant Co-Evolutionary Set (DCES) problem A. The input to the DCES problem is a phylogenetic forest and a target genome (red) corresponding to a certain leaf in all the phylogenetic trees (the leftmost leaf in this example). The goal is to find a subset of proteins in the target genome such that given this subset of proteins (and the phylogenetic forest), it will be possible to infer the rest of the genome; in this example, proteins from gene families T1, T2, T3, T4, T6, T8 are used to infer the rest of the target genome (the proteins marked with ′?′; proteins T5, T7, T9 in the example). B. The problem can be reduced to a version of the dominant set problem  (see details in section ’Algorithm for the Dominant co-evolutionary set problem’). In the reduction we build a graph that includes a node for each gene family and pairs of gene families are connected with edges if they have a strong co-evolutionary relation; we want to find a dominant set in this graph.