  A recent paper describes a framework for studying the computational
complexity of graph problems on monotone classes, that is those omitting a set
of graphs as a subgraph. If the problems lie in the framework, and many do,
then the computational complexity can be described for all monotone classes
defined by a finite set of omitted subgraphs. 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.
  The original framework had examples where hard versus easy were NP-complete
versus P, or at least quadratic versus almost linear. We give the first example
of a problem in the framework such that hardness is in the polynomial hierarchy
above NP. Considering a variant of the colouring game as studied by Bodlaender,
we show that with the restriction of bounded alternation, the list version of
this problem is contained in the framework. The hard cases are
$\Pi_{2k}^\mathrm{P}$-complete and the easy cases are in P.
  The cases in P comprise those classes for which the pathwidth is bounded.
Bodlaender explains that Sequential $3$-Colouring Construction Game is in P on
classes with bounded vertex separation number, which coincides with bounded
pathwidth on unordered graphs. However, these graphs are ordered with a playing
order for the two players, which corresponds to a prefix pattern in a
quantified formula. We prove that Sequential $3$-Colouring Construction Game is
Pspace-complete on some class of bounded pathwidth, using a celebrated result
of Atserias and Oliva.
  We consider several locally constrained variants of the homomorphism problem.
Like $C_5$-Colouring, none of these is in the framework. However, when we
consider the bounded-degree restrictions, we prove that each of these problems
is in our framework.
