  Graph monoids arise naturally in the study of non-stable K-theory of graph
C*-algebras and Leavitt path algebras. They play also an important role in the
current approaches to the realization problem for von Neumann regular rings. In
this paper, we characterize when a finitely generated conical refinement monoid
can be represented as a graph monoid. The characterization is expressed in
terms of the behavior of the structural maps of the associated $I$-system at
the free primes of the monoid.
