Geodesic
On maximal finite antichains in the homomorphism order of directed graphs
Discussiones Mathematicae. Graph Theory, Tome 23 (2003) no. 2, pp. 325-332

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We show that the pairs T,D_T where T is a tree and D_T its dual are the only maximal antichains of size 2 in the category of directed graphs endowed with its natural homomorphism ordering.
Keywords: chromatic number, homomorphism duality 
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Nesetril, Jaroslav; Tardif, Claude. On maximal finite antichains in the homomorphism order of directed graphs. Discussiones Mathematicae. Graph Theory, Tome 23 (2003) no. 2, pp. 325-332. http://geodesic.mathdoc.fr/item/DMGT_2003_23_2_a8/