The challenges of unbounded treewidth in parameterised subgraph counting problems

Kitty Meeks
2016 Discrete Applied Mathematics  
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
more » ... old, 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. 1 This result has previously been used along the same lines in different settings by a number of authors; a discussion of the relationship of our methods to previous work is given in Section 3.2.5. 2 This generalises a result of Curticapean and Marx [16] , published since the writing of an initial version of this paper, which proves a special case of this result. 3 For a stand-alone survey, the reader may omit Section 4.
doi:10.1016/j.dam.2015.06.019 fatcat:utjeznnsxfd45oozdipbxe5xyu