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A graph is called a pseudoforest if none of its connected components contains more than one cycle. A graph is an apex-pseudoforest if it can become a pseudoforest by removing one of its vertices. We identify 33 graphs that form the minor-obstruction set of the class of apexpseudoforests, i.e., the set of all minor-minimal graphs that are not apex-pseudoforests.

A graph is sub-unicyclic if it contains at most one cycle. We also say that 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, i.e., the set of all minor minimal graphs that do not belong in this class. For bigger values of k, we give an exact structural characterization of all the cactus graphs that are minor-obstructions of k-apex subunicyclic graphs and we enumerate them. This implies that, for every k, the class of k-apex sub-unicyclic graphs has at least 0.34 • k −2.5 (6.278) k minor-obstructions.