The challenges of unbounded treewidth in parameterised subgraph counting problems

Many of the existing tractability results for parameterised problems which involve finding or counting subgraphs with particular properties rely on bounding the treewidth of the minimal subgraphs having the desired property. In this paper, we give a number of hardness results – for decision, approximate counting and exact counting – in the case that this condition on the minimal subgraphs having the desired property does not hold. These results demonstrate that in some cases the bounded treewidth condition is necessary for the existence of an efficient algorithm, and lead to two dichotomies for problems which involve finding or counting multicolour subgraphs.