  A bipartite graph B is called a brace if it is connected and every matching
of size at most two in B is contained in some perfect matching of B and a cycle
C in B is called conformal if B-V(C) has a perfect matching. We show that there
do not exist two disjoint alternating paths that form a cross over a conformal
cycle C in a brace B if and only if one can reduce B, by an application of a
matching theoretic analogue of small clique sums, to a planar brace H in which
C bounds a face. We then utilise this result and provide a polynomial time
algorithm which solves the 2-linkage problem for alternating paths in bipartite
graphs with perfect matchings.
