Aspects of structural combinatorics - (Graph homomorphisms and their use)

This paper is based on a course delivered by the author at NCTS, National Chiao Tung University,Taiwan in February 1999. We survey results related to structural aspects of graph homomorphism. Our aim is to demonstrate that this forms today a compact collection of results and methods which perhaps deserve its name : structural combinatorics. Due to space limitations we concentrate on a sample of areas only: representation of algebraic structures by combinatorial ones (graphs), the poset of colour classes and corresponding algorithmic questions which lead to homomorphism dualities,blending algebraic and complexity approaches.