  The investigation of combinatorial diameters of polyhedra is a classical
topic in linear programming due to its connection with the possibility of an
efficient pivot rule for the simplex method. We are interested in the diameters
of polyhedra formed from the so-called parallel or series connection of
oriented matroids: oriented matroids are the natural way to connect
representable matroid theory with the combinatorics of linear programming, and
these connections are fundamental operations for the construction of more
complicated matroids from elementary matroid blocks.
  We prove that, for polyhedra whose combinatorial diameter satisfies the
Hirsch-conjecture bound regardless of the right-hand sides in a standard-form
description, the diameters of their parallel or series connections remain small
in the Hirsch-conjecture bound. These results are a substantial step toward
devising a diameter bound for all polyhedra defined through totally-unimodular
matrices based on Seymour's famous decomposition theorem.
  Our proof techniques and results exhibit a number of interesting features.
While the parallel connection leads to a bound that adds just a constant, for
the series connection one has to linearly take into account the maximal value
in a specific coordinate of any vertex. Our proofs also require a careful
treatment of non-revisiting edge walks in degenerate polyhedra, as well as the
construction of edge walks that may take a `detour' to facets that satisfy the
non-revisiting conjecture when the underlying polyhedron may not.
