  We introduce a new combinatorial abstraction for the graphs of polyhedra. The
new abstraction is a flexible framework defined by combinatorial properties,
with each collection of properties taken providing a variant for studying the
diameters of polyhedral graphs. One particular variant has a diameter which
satisfies the best known upper bound on the diameters of polyhedra. Another
variant has superlinear asymptotic diameter, and together with some
combinatorial operations, gives a concrete approach for disproving the Linear
Hirsch Conjecture.
