Abstract

Tutte introduced a V by V skew--symmetric matrix T = (t ij ), called the Tutte matrix, associated with a simple graph G = (V; E). He associates an indeterminate z e with each e 2 E, then defines t ij = \Sigmaz e when ij = e 2 E, and t ij = 0 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. 1 Introduction Let G = (V; E) be a simple graph, and let (z e : e 2 E) be algebraically independent commuting indeterminates. We define a V by V skew--symmetric matrix T = (t ij ), called the Tutte matrix of G, such that t ij = \Sigmaz e if ij = e 2 E, and t ij = 0 otherwise. Tutte observed that T is nonsingular (that is, its determinant is not...

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