Two Disjoint Alternating Paths in Bipartite Graphs
Combinatorics
2021-10-06 v1 Discrete Mathematics
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.
Keywords
Cite
@article{arxiv.2110.02013,
title = {Two Disjoint Alternating Paths in Bipartite Graphs},
author = {Archontia C. Giannopoulou and Sebastian Wiederrecht},
journal= {arXiv preprint arXiv:2110.02013},
year = {2021}
}