Minimal digraph obstructions for small matrices

TL;DR

This paper classifies all minimal digraph obstructions for small 2x2 matrices, providing a complete list of digraphs that cannot be partitioned according to these matrices, with implications for digraph partitioning problems.

Contribution

It provides the first complete enumeration of minimal obstructions for all 2x2 matrices in the context of digraph partitioning.

Findings

List of all minimal M-obstructions for every 2x2 matrix M

Complete characterization of non-partitionable digraphs for small matrices

Foundation for future research in digraph partitioning problems

Abstract

Given a $\{ 0, 1, \ast \}$-matrix $M$, a minimal $M$-obstruction is a digraph $D$ such that $D$ is not $M$-partitionable, but every proper induced subdigraph of $D$ is. In this note we present a list of all the $M$-obstructions for every $2 \times 2$ matrix $M$. Notice that this note will be part of a larger paper, but we are archiving it now so we can cite the results.