  Provan and Billera introduced notions of (weak) decomposability of simplicial
complexes as a means of attempting to prove polynomial upper bounds on the
diameter of the facet-ridge graph of a simplicial polytope. Recently, De Loera
and Klee provided the first examples of simplicial polytopes that are not
weakly vertex-decomposable. These polytopes are polar to certain simple
transportation polytopes. In this paper, we refine their analysis to prove that
these $d$-dimensional polytopes are not even weakly $O(\sqrt{d})$-decomposable.
As a consequence, (weak) decomposability cannot be used to prove a polynomial
version of the Hirsch conjecture.
