Minimal digraph obstructions for small matrices
  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. 
Pavol Hell and C\'esar Hern\'andez-Cruz
math.CO