  The parameters of a linear compartment model are usually estimated from
experimental input-output data. A problem arises when infinitely many parameter
values can yield the same result; such a model is called unidentifiable. In
this case, one can search for an identifiable reparametrization of the model: a
map which reduces the number of parameters, such that the reduced model is
identifiable. We study a specific class of models which are known to be
unidentifiable. Using algebraic geometry and graph theory, we translate a
criterion given by Meshkat and Sullivant for the existence of an identifiable
scaling reparametrization to a new criterion based on the rank of a weighted
adjacency matrix of a certain bipartite graph. This allows us to derive several
new constructions to obtain graphs with an identifiable scaling
reparametrization. Using these constructions, a large subclass of such graphs
is obtained. Finally, we present a procedure of subdividing or deleting edges
to ensure that a model has an identifiable scaling reparametrization.
