  Circuits play a fundamental role in the theory of linear programming due to
their intimate connection to algorithms of combinatorial optimization and the
efficiency of the simplex method. We are interested in better understanding the
properties of circuit walks in integral polyhedra. In this paper, we introduce
a hierarchy for integral polyhedra based on different types of behavior
exhibited by their circuit walks. Many problems in combinatorial optimization
fall into the most interesting categories of this hierarchy - steps of circuit
walks only stop at integer points, at vertices, or follow actual edges. We
classify several classical families of polyhedra within the hierarchy,
including 0/1-polytopes, polyhedra defined by totally unimodular matrices, and
more specifically matroid polytopes, transportation polytopes, and partition
polytopes. Finally, we prove three characterizations of the simple polytopes
that appear in the bottom level of the hierarchy where all circuit walks are
edge walks, showing that such polytopes constitute a generalization of
simplices and parallelotopes.
