Minor-obstructions for apex sub-unicyclic graphs

Abstract A graph is sub-unicyclic if it contains at most one cycle. A graph G is k -apex sub-unicyclic if it can become sub-unicyclic by removing k of its vertices. We identify 29 graphs that are the minor-obstructions of the class of 1-apex sub-unicyclic graphs. For bigger values of k , we give an exact structural characterization of all the cactus graphs that are minor-obstructions of k -apex sub-unicyclic graphs and we enumerate them. This implies that, for k big enough, the class of k -apex sub-unicyclic graphs has at least 0 . 33 ⋅ k − 2 . 5 ( 6 . 278 ) k + 1 minor-obstructions.