Minor-obstructions for apex sub-unicyclic graphs

A graph is if it contains at most one cycle. A graph is if it can become sub-unicyclic by removing 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 , we give an exact structural characterization of all the cactus graphs that are minor-obstructions of -apex sub-unicyclic graphs and we enumerate them. This implies that, for big enough, the class of -apex sub-unicyclic graphs has at least minor-obstructions.