Matroid Matching Via Mixed Skew-Symmetric Matrices

Tutte associates a V by V skew-symmetric matrix T, having indeterminate entries, with a graph G=(V,E). This matrix, called the Tutte matrix, has rank exactly twice the size of a maximum cardinality matching of G. Thus, to find the size of a maximum matching it suffices to compute the rank of T. We consider the more general problem of computing the rank of T + K where K is a real V by V skew-symmetric matrix. This modest generalization of the matching problem contains the linear matroid matching problem and, more generally, the linear delta-matroid parity problem. We present a tight upper bound on the rank of T + K by decomposing T + K into a sum of matrices whose ranks are easy to compute.

This is a preview of subscription content, access via your institution.

Author information



Corresponding author

Correspondence to James Geelen.

Additional information

Part of this research was done while the authors visited the Fields Institute in Toronto, Canada. The research was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Geelen, J., Iwata, S. Matroid Matching Via Mixed Skew-Symmetric Matrices. Combinatorica 25, 187–215 (2005).

Download citation

Mathematics Subject Classification (2000):

  • 05C70