An Algebraic Matching Algorithm

V

by V skew-symmetric matrix , called the Tutte matrix, associated with a simple graph G=(V,E). He associates an indeterminate with each , then defines when , and otherwise. The rank of the Tutte matrix is exactly twice the size of a maximum matching of G. Using linear algebra and ideas from the Gallai–Edmonds decomposition, we describe a very simple yet efficient algorithm that replaces the indeterminates with constants without losing rank. Hence, by computing the rank of the resulting matrix, we can efficiently compute the size of a maximum matching of a graph.