3,313 research outputs found
On infinite-finite duality pairs of directed graphs
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
Caterpillar dualities and regular languages
We characterize obstruction sets in caterpillar dualities in terms of regular
languages, and give a construction of the dual of a regular family of
caterpillars. We show that these duals correspond to the constraint
satisfaction problems definable by a monadic linear Datalog program with at
most one EDB per rule
Dualities and dual pairs in Heyting algebras
We extract the abstract core of finite homomorphism dualities using the
techniques of Heyting algebras and (combinatorial) categories.Comment: 17 pages; v2: minor correction
On First-Order Definable Colorings
We address the problem of characterizing -coloring problems that are
first-order definable on a fixed class of relational structures. In this
context, we give several characterizations of a homomorphism dualities arising
in a class of structure
Regular families of forests, antichains and duality pairs of relational structures
Homomorphism duality pairs play crucial role in the theory of 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 we characterize the infinite-finite antichain
dualities and infinite-finite dualities with trees or forest on the left hand
side. This work builds on our earlier papers that gave several examples of
infinite-finite antichain duality pairs of directed graphs and a complete
characterization for caterpillar dualities
On the order of countable graphs
A set of graphs is said to be independent if there is no homomorphism between
distinct graphs from the set. We consider the existence problems related to the
independent sets of countable graphs. While the maximal size of an independent
set of countable graphs is 2^omega the On Line problem of extending an
independent set to a larger independent set is much harder. We prove here that
singletons can be extended (``partnership theorem''). While this is the best
possible in general, we give structural conditions which guarantee independent
extensions of larger independent sets. This is related to universal graphs,
rigid graphs and to the density problem for countable graphs
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