  Parameterised subgraph counting problems are the most thoroughly studied
topic in the theory of parameterised counting, and there has been significant
recent progress in this area. Many of the existing tractability results for
parameterised problems which involve finding or counting subgraphs with
particular properties rely on bounding the treewidth of these subgraphs in some
sense; here, we prove a number of hardness results for the situation in which
this bounded treewidth condition does not hold, resulting in dichotomies for
some special cases of the general subgraph counting problem. The paper also
gives a thorough survey of known results on this subject and the methods used,
as well as discussing the relationships both between multicolour and uncoloured
versions of subgraph counting problems, and between exact counting, approximate
counting and the corresponding decision problems.
