Many facets of dualities

Abstract

In this paper we survey results related to homomorphism dualities for graphs, and more generally, for finite structures. This is related to some of the classical combinatorial problems, such as colorings of graphs and hypergraphs, and also to recently intensively studied Constraint Satisfaction Problems. On the other side dualities are related to the descriptive complexity and First Order definability as well as to universal graphs. And in yet another context they can be expressed as properties of the homomorphism order of structures. In the contemporary context homomorphism dualities are a complex area and it is our aim to describe some of the main ideas only. However we introduce the four conceptually different proofs of the existence of duals thus indicating the versatility of this notion. Particularly we describe setting of restricted dualities and the role of bounded expansion classes. © 2009 Springer Berlin Heidelberg.