  We introduce a class of linear compartmental models called identifiable
path/cycle models which have the property that all of the monomial functions of
parameters associated to the directed cycles and paths from input compartments
to output compartments are identifiable and give sufficient conditions to
obtain one. Removing leaks, we then show how one can obtain a locally
identifiable model from an identifiable path/cycle model. These identifiable
path/cycle models yield the only identifiable models with certain conditions on
their graph structure and thus we provide necessary and sufficient conditions
for identifiable models with certain graph properties. A sufficient condition
based on the graph structure of the model is also provided so that one can test
if a model is an identifiable path/cycle model by examining the graph itself.
We also provide some necessary conditions for identifiability based on graph
structure. Our proofs use algebraic and combinatorial techniques.
