A tight bound on the length of odd cycles in the incompatibility graph of a non-C1P matrix

Authors:
Mehrnoush Malekesmaeili;Cedric Chauve;Tamon Stephen
Affiliations:
Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada;Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada;Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada
Venue:
Information Processing Letters
Year:
2012

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Abstract

A binary matrix has the Consecutive Ones Property (C1P) if it is possible to order the columns so that all 1s are consecutive in every row. In [McConnell, SODA 2004, pp. 768-777] the notion of incompatibility graph of a binary matrix was introduced and it was shown that odd cycles of this graph provide a certificate that a matrix does not have the Consecutive Ones Property. A bound of k+2 was claimed for the smallest odd cycle of a non-C1P matrix with k columns. In this Note we show that this result can be obtained simply and directly via Tucker patterns, and that the correct bound is k+2 when k is even, but k+3 when k is odd.