Minimal Separating Sets in Surfaces

Given a connected topolgical space X, we say that L ⊆ X is a minimal separating set if removing L from X gives a disconnected surface, butremoving any proper subset of L leaves the surface connected. We classify which embeddings of topological graphs are minimal separating in an orientable surface X with genus g, and construct a computer program to compute the number of such embeddings, and the number of topological graphs which admit such an embedding for g ≤ 5.