Thick Forests

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 we show to be the largest class of perfect thick graphs. Recognising membership of a class of thick graphs is NP-complete unless the class of thin graphs is triangle-free, so we focus on this case. Even then membership can be NP-complete. However, we show that the class of thick forests can be recognised in polynomial time. We consider two well-studied combinatorial problems on thick graphs, independent sets and proper colourings. Since determining the independence or chromatic number of a perfect graph is known to be tractable, we examine the complexity of counting all independent sets and colourings in thick forests. Finally, we consider two parametric extensions to larger classes of thick graphs: where the parameter is the size of the thin graph, and where the parameter is its treewidth.