  This paper develops a fundamental theory of realizations of linear and group
codes on general graphs using elementary group theory, including basic group
duality theory. Principal new and extended results include: normal realization
duality; analysis of systems-theoretic properties of fragments of realizations
and their connections; "minimal = trim and proper" theorem for cycle-free
codes; results showing that all constraint codes except interface nodes may be
assumed to be trim and proper, and that the interesting part of a cyclic
realization is its "2-core;" notions of observability and controllability for
fragments, and related tests; relations between state-trimness and
controllability, and dual state-trimness and observability.
