On the structure of matrices avoiding interval-minor patterns

Abstract

We study the structure of 01-matrices avoiding a pattern P as an interval minor. We focus on critical P -avoiders, i.e., on the P -avoiding matrices in which changing a 0-entry to a 1-entry always creates a copy of P as an interval minor.Let Q be the 3 × 3 permutation matrix corresponding to the permutation 231. As our main result, we show that for every pattern P that has no rotated copy of Q as interval minor, there is a constant cP such that any row and any column in any critical P -avoiding matrix can be partitioned into at most cP intervals, each consisting entirely of 0-entries or entirely of 1-entries. In contrast, for any pattern P that contains a rotated copy of Q, we construct critical P -avoiding matrices of arbitrary size n × n having a row with Ω(n) alternating intervals of 0-entries and 1-entries.