  We show that for various classes C of sparse graphs, and several measures of
distance to such classes (such as edit distance and elimination distance), the
problem of determining the distance of a given graph G to C is fixed-parameter
tractable. The results are based on two general techniques. The first of these,
building on recent work of Grohe et al. establishes that any class of graphs
that is slicewise nowhere dense and slicewise first-order definable is FPT. The
second shows that determining the elimination distance of a graph G to a
minor-closed class C is FPT.
