### Abstract

The (A,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 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.

Original language | English |
---|---|

Pages (from-to) | 807-819 |

Number of pages | 13 |

Journal | Order |

Volume | 30 |

Issue number | 3 |

DOIs | |

Publication status | Published - Nov 2013 |

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### Keywords

- Constraint satisfaction problem
- Duality pairs
- General relational structures
- Graph homomorphism
- Nondeterministic finite automaton
- Regular languages

### ASJC Scopus subject areas

- Algebra and Number Theory
- Geometry and Topology
- Computational Theory and Mathematics

### Cite this

*Order*,

*30*(3), 807-819. https://doi.org/10.1007/s11083-012-9278-9

**On Infinite-finite Duality Pairs of Directed Graphs.** / Erdos, Péter L.; Tardif, Claude; Tardos, G.

Research output: Contribution to journal › Article

*Order*, vol. 30, no. 3, pp. 807-819. https://doi.org/10.1007/s11083-012-9278-9

}

TY - JOUR

T1 - On Infinite-finite Duality Pairs of Directed Graphs

AU - Erdos, Péter L.

AU - Tardif, Claude

AU - Tardos, G.

PY - 2013/11

Y1 - 2013/11

N2 - The (A,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 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.

AB - The (A,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 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.

KW - Constraint satisfaction problem

KW - Duality pairs

KW - General relational structures

KW - Graph homomorphism

KW - Nondeterministic finite automaton

KW - Regular languages

UR - http://www.scopus.com/inward/record.url?scp=84885674670&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84885674670&partnerID=8YFLogxK

U2 - 10.1007/s11083-012-9278-9

DO - 10.1007/s11083-012-9278-9

M3 - Article

AN - SCOPUS:84885674670

VL - 30

SP - 807

EP - 819

JO - Order

JF - Order

SN - 0167-8094

IS - 3

ER -