  We introduce a class of parameterised counting problems on graphs, p-#Induced
Subgraph With Property(\Phi), which generalises a number of problems which have
previously been studied. This paper focusses on the case in which \Phi defines
a family of graphs whose edge-minimal elements all have bounded treewidth; this
includes the special case in which \Phi describes the property of being
connected. We show that exactly counting the number of connected induced
k-vertex subgraphs in an n-vertex graph is #W[1]-hard, but on the other hand
there exists an FPTRAS for the problem; more generally, we show that there
exists an FPTRAS for p-#Induced Subgraph With Property(\Phi) whenever \Phi is
monotone and all the minimal graphs satisfying \Phi have bounded treewidth. We
then apply these results to a counting version of the Graph Motif problem.

Computational Complexity;Combinatorics