Minor-Obstructions for Apex Sub-unicyclic Graphs.

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.