Abstract 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. A conformal cross over some cycle C is a pair of disjoint paths P₁ P 1, P₂ P 2 which are internally disjoint from C, the endpoints of each path separate the endpoints of the other path on C, and both C P₁ P₂ C ∪ P 1 ∪ P 2 and B- (V (C) V (P₁) V (P₂) ) B - (V (C) ∪ V (P 1) ∪ V (P 2) ) have a perfect matching. We show that if C is a 4-cycle in a brace B, then C has a a conformal cross if and only if B contains K₃, ₃ K 3, 3 as a matching minor. This result implies a polynomial time algorithm which solves the 2-linkage problem for alternating paths in bipartite graphs with perfect matchings.
Giannopoulou et al. (2026) studied this question.