Point determining digraphs, $\{0,1\}$-matrix partitions, and dualities in full homomorphisms

TL;DR

This paper proves a property of point-determining digraphs and applies it to bound the size of minimal obstructions for certain matrix partitions, extending previous results from undirected graphs to directed graphs.

Contribution

It introduces a new vertex-removal property for point-determining digraphs and extends bounds on minimal obstructions from undirected graphs to directed graphs and general matrices.

Findings

Existence of a vertex in point-determining digraphs whose removal preserves point-determining property.

Bound on the size of minimal $M$-obstructions for $oxed{0,1}$-matrices with specified diagonal zeros and ones.

Extension of previous undirected graph results to digraphs and general matrices.

Abstract

We prove that every point-determining digraph $D$ contains a vertex $v$ such that $D-v$ is also point determining. We apply this result to show that for any $\{0,1\}$-matrix $M$, with $k$ diagonal zeros and $\ell$ diagonal ones, the size of a minimal $M$-obstruction is at most $(k+1)(\ell+1)$. This extends the results of Sumner, and of Feder and Hell, from undirected graphs and symmetric matrices to digraphs and general matrices.