Abstract
We fully characterise the situations where the existence of a homomorphism from a digraph G to at least one of a finite set \(\mathcal{H}\) of directed graphs is determined by a finite number of forbidden subgraphs. We prove that these situations, called generalised dualities, are characterised by the non-existence of a homomorphism to G from a finite set of forests.
Furthermore, we characterise all finite maximal antichains in the partial order of directed graphs ordered by the existence of homomorphism. We show that these antichains correspond exactly to the generalised dualities. This solves a problem posed in [1]. Finally, we show that it is NP-hard to decide whether a finite set of digraphs forms a maximal antichain.
This research was partially supported by the eu network combstru.
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Foniok, J., Nešetřil, J., Tardif, C. (2006). Generalised Dualities and Finite Maximal Antichains. In: Fomin, F.V. (eds) Graph-Theoretic Concepts in Computer Science. WG 2006. Lecture Notes in Computer Science, vol 4271. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11917496_3
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DOI: https://doi.org/10.1007/11917496_3
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