Two Disjoint Alternating Paths in Bipartite Graphs

TL;DR

This paper characterizes the non-existence of two disjoint alternating paths crossing a conformal cycle in bipartite braces and provides a polynomial algorithm for the 2-linkage problem in such graphs.

Contribution

It introduces a structural characterization of braces related to alternating paths and develops a polynomial-time algorithm for the 2-linkage problem in bipartite graphs.

Findings

Characterization of braces with no two disjoint alternating paths crossing a conformal cycle.

Reduction of braces to planar braces using matching theoretic small clique sums.

Polynomial-time algorithm for the 2-linkage problem in bipartite graphs with perfect matchings.

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 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.