Abstract
A Ck-free 2-matching in an undirected graph is a simple 2-matching which does not contain cycles of length k or less. The complexity of finding the maximum Ck-free 2-matching in a given graph varies depending on k and the type of input graph. The case where k=4 and the graph is bipartite, which is called the maximum square-free 2-matching problem in bipartite graphs, is well-solved. Previous results for this special case include min-max theorems, polynomial combinatorial algorithms, a linear programming formulation with dual integrality for a weighted version, and discrete convex structures.In this paper, we further investigate the structure of square-free 2-matchings in bipartite graphs and present new decomposition theorems. These theorems serve as analogues of the Dulmage-Mendelsohn decomposition for matchings in bipartite graphs and the Edmonds-Gallai decomposition for matchings in nonbipartite graphs. We exhibit two canonical minimizers for the set function in the min-max formula, and a characterization of the maximum square-free 2-matchings with the aid of these canonical minimizers.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
