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

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+2k+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+2k+2 when k   is even, but k+3k+3 when k is odd.

► Consecutive Ones Property (C1P).
► Certificates of non-C1P.
► Characterization of minimal certificates.