An FPT certifying algorithm for the vertex-deletion problem

We provide a fixed-parameter certifying algorithm for the Vertex-deletion problem. An upper bound for the size of the forbidden set for C+kv is shown demonstrating that, for all hereditary classes characterised by a finite forbidden set, the class C+kv is characterised by a finite forbidden set. The certifying algorithm runs in time O(f(k)·n) and can be verified in O(n) in the affirmative case and O(f(k)) in the negative case. This is the first known fixed-parameter certifying algorithm, as far as the authors are aware.