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We prove that every point-determining digraph $D$ contains a vertex $v$ suchthat $D-v$ 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, thesize of a minimal $M$-obstruction is at most $(k+1)(\ell+1)$. This extends theresults of Sumner, and of Feder and Hell, from undirected graphs and symmetricmatrices to digraphs and general matrices.

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