On the structure of matrices avoiding interval-minor patterns

TL;DR

This paper investigates the structure of 01-matrices avoiding certain interval-minor patterns, revealing a dichotomy based on the presence of a specific permutation pattern, with implications for matrix partitioning.

Contribution

It characterizes the structure of critical P-avoiding matrices, establishing bounds on row and column partitions for patterns without a rotated 231 permutation minor.

Findings

For patterns without a rotated 231 minor, matrices can be partitioned into a bounded number of uniform intervals.

For patterns containing a rotated 231 minor, matrices can have rows with arbitrarily many alternating intervals.

The structural difference hinges on the presence or absence of a rotated 231 permutation minor.

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 3x3 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 c(P) such that any row and any column in any critical P-avoiding matrix can be partitioned into at most c(P) 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 $n\times n$ having a row with $\Omega(n)$ alternating intervals of 0-entries and 1-entries.