Graph Homomorphism, Monotone Classes and Bounded Pathwidth

Monotone graph classes are those described by some set of forbidden subgraphs, i.e. H-subgraph-free for some set of graphs H. In recent work a framework was described for the study of such graph classes, if a problem falls into the framework then its computational complexity can be described, for all graph classes defined by a finite set of omitted subgraphs. This allows a dichotomy to be specified between those classes for which the problem is hard and easy. Here we consider several variants of the homomorphism problem in relation to this framework. It is known that certain homomorphism problems, e.g. C-5-Colouring, do not sit in the framework. By contrast, we show that the more general problem of GRAPH HOMOMORPHISM does sit in the framework. We also give the first example of a problem in the framework such that hardness is in the polynomial hierarchy above NP. This comes from a list colouring game, where we show that with the restriction of bounded alternation, this problem is contained in the framework. The hard cases are. Pi(P)(2k)-complete and the easy cases are in P. Finally we consider several locally constrained variants of the homomorphism problem, namely the locally bijective, surjective and injective variants. Like C-5-Colouring, none of these is in the framework. However, where a bounded-degree restrictions are considered, we prove that each of these problems is in our framework.