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Structural properties and decomposition of linear balanced matrices. (English) Zbl 0767.90068
Summary: Claude Berge defines a (0,1) matrix \(A\) to be linear if \(A\) does not contain a \(2\times 2\) submatrix of all ones. A \((0,1)\) matrix \(A\) is balanced if \(A\) does not contain a square submatrix of odd order with two ones per row and column. The contraction of a row \(i\) of a matrix consists of the removal of row \(i\) and all the columns that have a 1 in the entry corresponding to row \(i\).
We show that if a linear balanced matrix \(A\) does not belong to a subclass of totally unimodular matrices, then \(A\) or \(A^ T\) contains a row \(i\) such that the submatrix obtained by contracting row \(i\) has a block-diagonal structure.

MSC:
90C27 Combinatorial optimization
52B12 Special polytopes (linear programming, centrally symmetric, etc.)
90C35 Programming involving graphs or networks
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