Point determining digraphs, [formula omitted]-matrix partitions, and dualities in full homomorphisms

Abstract

A digraph D is point determining if for any two distinct vertices u,v there exists a vertex w which has an arc to (or from) exactly one of u,v. 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 ℓ diagonal ones, the size of a minimal M-obstruction is at most (k+1)(ℓ+1). This is a best possible bound, and it extends the results of Sumner, and of Feder and Hell, from undirected graphs and symmetric matrices to digraphs and general matrices.