Abstract
A graph is {\em sub-unicyclic} if it contains at most one cycle. A graph $G$ is {\em $k$-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 $0.33\cdot k^{-2.5}(6.278)^{k+1}$ minor-obstructions.