Outerplanar obstructions for a feedback vertex set

For k≥1, let Fk be the class of graphs that contain k vertices meeting all its cycles. The minor-obstruction set for Fk is the set obs(Fk) containing all minor-minimal graphs that do not belong to Fk. We denote by Yk the set of all outerplanar graphs in obs(Fk). In this paper, we provide a precise characterization of the class Yk. Then, using singularity analysis over the counting series obtained with the Symbolic Method, we prove that ∣Yk∣∼C′⋅k−5/2⋅ρ−k where C′≐0.02575057 and ρ−1≐14.49381704 (ρ is the smallest positive root of a quadratic equation).