  We consider classes of graphs, which we call thick graphs, that have the
vertices of a corresponding thin graph replaced by cliques and the edges
replaced by cobipartite graphs. In particular, we consider the case of thick
forests, which are a class of perfect graphs. Whereas recognising membership of
most classes of thick graphs is NP-complete, we show that thick forests can be
recognised in polynomial time. We consider two well-studied combinatorial
problems on thick graphs, independent sets and proper colourings. Since finding
the independence or chromatic number of a perfect graph is well known to be in
polynomial time, we examine the complexity of counting all independent sets and
colourings in a thick forest. Finally, we consider two extensions of our
results to larger classes: thick triangle-free graphs and thick
bounded-treewidth graphs.
