  We generalize a result on the Heisenberg Lie algebra that gives restrictions
to possible Lie bialgebra cobrackets on 2-step nilpotent algebras with some
additional properties. For the class of 2-step nilpotent Lie algebras coming
from graphs, we describe these extra properties in a very easy
graph-combinatorial way. We exhibit applications for $\mathfrak f_n$, the free
2-step nilpotent Lie algebra.
