We study the class of all finite directed graphs (digraphs) up to primitive positive constructibility. The resulting order has a unique maximal element, namely the digraph P1 with one vertex and no edges. The digraph P1 has a unique maximal lower bound, namely the digraph P2 with two vertices and one directed edge. Our main result is a complete description of the maximal lower bounds of P2; we call these digraphs submaximal. We show that every digraph that is not equivalent to P1 and P2 is below one of the submaximal digraphs.
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The authors would like to thank the anonymous referees for thoroughly reading our article as well as giving helpful comments that improved the final article.
The second author is supported by DFG Graduiertenkolleg 1763 (QuantLA).
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Bodirsky, M., Starke, F. Maximal Digraphs with Respect to Primitive Positive Constructability. Combinatorica 42 (Suppl 1), 997–1010 (2022). https://doi.org/10.1007/s00493-022-4918-1
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