  We consider, for complete bipartite graphs, the convex hulls of
characteristic vectors of all matchings, extended by a binary entry indicating
whether the matching contains two specific edges. These polytopes are
associated to the quadratic matching problems with a single linearized
quadratic term. We provide a complete irredundant inequality description, which
settles a conjecture by Klein (Ph.D. thesis, TU Dortmund, 2015). In addition,
we also derive facetness and separation results for the polytopes. The
completeness proof is based on a geometric relationship to a matching polytope
of a nonbipartite graph. Using standard techniques, we finally extend the
result to capacitated b-matchings.
