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.
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