Before revision:  Here {\sf poly} is a polynomial function whose degree depends on the maximum size of a minor-obstruction of {\cal G} , i. e., the minor-minimal set of graphs not belonging to G . In the special case where {\cal G} excludes some apex graph as a minor, we give an alternative algorithm running in 2^{{\sf poly}(k)}\cdot n^2 time .
After revision:  Here {\sf poly} is a polynomial function whose degree depends on the maximum size of a minor-obstruction of {\cal G} , i. e., the minor-minimal set of graphs not belonging to G . In the special case where {\cal G} excludes some apex graph as a minor, we give an alternative algorithm running in 2^{{\sf poly}(k)}\cdot n^2 -time .