On infinite-finite duality pairs of directed graphs

TL;DR

This paper initiates the study of infinite-finite duality pairs in directed graphs, establishing foundational properties and examples, and revealing the existence of such dualities where previously unknown.

Contribution

It provides the first detailed analysis of infinite-finite duality pairs in directed graphs, including bounds, properties, and explicit examples.

Findings

Elements of A are forests if A is an antichain.

Constructed examples include paths and trees.

Infinite-finite antichain dualities can exist, contrary to previous beliefs.

Abstract

The (A,D) duality pairs play crucial role in the theory of general relational structures and in the Constraint Satisfaction Problem. The case where both classes are finite is fully characterized. The case when both side 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 D, and show that the elements of A must be equivalent to forests if A is an antichain. Then we construct instructive examples, where the elements of A are paths or trees. Note that the existence of infinite-finite antichain dualities was not previously known.