Outerplanar obstructions for a feedback vertex set

For k ≥ 1 , let F k be the class of graphs that contain k vertices meeting all its cycles. The minor-obstruction set for F k is the set obs ( F k ) containing all minor-minimal graphs that do not belong to F k . We denote by Y k the set of all outerplanar graphs in obs ( F k ) . In this paper, we provide a precise characterization of the class Y k . Then, using singularity analysis over the counting series obtained with the Symbolic Method, we prove that ∣ Y k ∣ ∼ 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).