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

On First-Order Definable Colorings

We address the problem of characterizing $H$-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

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 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

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

Homomorphisms and Structural Properties of Relational Systems

Two main topics are considered: The characterisation of finite homomorphism dualities for relational structures, and the splitting property of maximal antichains in the homomorphism order.Comment: PhD Thesis, 77 pages, 14 figure

On Ramsey properties of classes with forbidden trees

Let F be a set of relational trees and let Forbh(F) be the class of all structures that admit no homomorphism from any tree in F; all this happens over a fixed finite relational signature $\sigma$. There is a natural way to expand Forbh(F) by unary relations to an amalgamation class. This expanded class, enhanced with a linear ordering, has the Ramsey property.Comment: Keywords: forbidden substructure; amalgamation; Ramsey class; partite method v2: changed definition of expanded class; v3: final versio

Discrete Convex Functions on Graphs and Their Algorithmic Applications

The present article is an exposition of a theory of discrete convex functions on certain graph structures, developed by the author in recent years. This theory is a spin-off of discrete convex analysis by Murota, and is motivated by combinatorial dualities in multiflow problems and the complexity classification of facility location problems on graphs. We outline the theory and algorithmic applications in combinatorial optimization problems

Dualities in full homomorphisms

AbstractIn this paper we study dualities of graphs and, more generally, relational structures with respect to full homomorphisms, that is, mappings that are both edge- and non-edge-preserving. The research was motivated, a.o., by results from logic (concerning first order definability) and Constraint Satisfaction Problems. We prove that for any finite set of objects B (finite relational structures) there is a finite duality with B to the left. It appears that the surprising richness of these dualities leads to interesting problems of Ramsey type; this is what we explicitly analyze in the simplest case of graphs