We show that the pairs where T is a tree and its dual are the only maximal antichains of size 2 in the category of directed graphs endowed with its natural homomorphism ordering.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1205, author = {Jaroslav Nesetril and Claude Tardif}, title = {On maximal finite antichains in the homomorphism order of directed graphs}, journal = {Discussiones Mathematicae Graph Theory}, volume = {23}, year = {2003}, pages = {325-332}, zbl = {1057.05036}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1205} }
Jaroslav Nesetril; Claude Tardif. On maximal finite antichains in the homomorphism order of directed graphs. Discussiones Mathematicae Graph Theory, Tome 23 (2003) pp. 325-332. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1205/
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