  This paper is concerned with the local reducibility properties of linear
realizations of codes on finite graphs.
  Trimness and properness are dual properties of constraint codes. A linear
realization is locally reducible if any constraint code is not both trim and
proper. On a finite cycle-free graph, a linear realization is minimal if and
only if every constraint code is both trim and proper.
  A linear realization is called observable if it is one-to-one, and
controllable if all constraints are independent. Observability and
controllability are dual properties. An unobservable or uncontrollable
realization is locally reducible. A parity-check realization is uncontrollable
if and only if it has redundant parity checks. A tail-biting trellis
realization is uncontrollable if and only if its trajectories partition into
disconnected subrealizations. General graphical realizations do not share this
property.
