Low degree minimal generators of phylogenetic semigroups

The phylogenetic semigroup on a graph generalizes the Jukes-Cantor binary model on a tree. Minimal generating sets of phylogenetic semigroups have been described for trivalent trees by Buczy\'nska and Wi\'sniewski, and for trivalent graphs with first Betti number 1 by Buczy\'nska. We characterize degree two minimal generators of the phylogenetic semigroup on any trivalent graph. Moreover, for any graph with first Betti number 1 and for any trivalent graph with first Betti number 2 we describe the minimal generating set of its phylogenetic semigroup.