  The study of the combinatorial diameter of a polyhedron is a classical topic
in linear-programming theory due to its close connection with the possibility
of a polynomial simplex-method pivot rule. The 2-sum operation is a classical
operation for graphs, matrices, and matroids; we extend this definition to
polyhedra. We analyze the diameters of 2-sum polyhedra, which are those
polyhedra that arise from this operation. These polyhedra appear in matroid and
integer-programming theory as a natural way to link two systems in a joint
model with a single shared constraint and the 2-sum also appears as a key
operation in Seymour's decomposition theorem for totally-unimodular matrices.
  We show that the diameter of a 2-sum polyhedron is quadratic in the diameters
of its summands. The methods transfer to a linear bound for the addition of a
unit column to an equality system, or equivalently, to the relaxation of an
equality constraint to an inequality constraint. Further, we use our methods to
analyze the distance between vertices on certain faces of a 3-sum polyhedron.
