, Volume 30, Issue 3, pp 807-819

On Infinite–finite Duality Pairs of Directed Graphs

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Abstract

The $({\mathcal{A}},{\mathcal{D}})$ duality pairs play a crucial role in the theory of general relational structures and in Constraint Satisfaction Problems. The case where both sides are finite is fully characterized. The case where both sides are infinite seems to be very complex. It is also known that no finite–infinite duality pair is possible if we make the additional restriction that both classes are antichains. In this paper (which is the first one of a series) we start the detailed study of the infinite–finite case. Here we concentrate on directed graphs. We prove some elementary properties of the infinite–finite duality pairs, including lower and upper bounds on the size of ${\mathcal{D}}$ , and show that the elements of ${\mathcal{A}}$ must be equivalent to forests if ${\mathcal{A}}$ is an antichain. Then we construct instructive examples, where the elements of ${\mathcal{A}}$ are paths or trees. Note that the existence of infinite–finite antichain dualities was not previously known.

Research of Péter L. Erdős was supported in part by the Hungarian NSF, under contracts NK 78439 and K 68262.
Research of Claude Tardif was supported by grants from NSERC and ARP.
Research of Gábor Tardos was supported in part by an NSERC Discovery grant and by the Hungarian OTKA grants T-046234, AT048826 and NK-62321.