  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.
