Classification of minimal separating sets of low genus surfaces

A minimal separating set in a connected topological space X is a subset $L\subset X$ with the property that $X\setminus L$ is disconnected, but if $L^{\prime }$ is a proper subset of L, then $X\setminus L^{\prime }$ is connected. Such sets appear in a variety of contexts. For example, in a wide class of metric spaces, if we choose distinct points p and q, then the set of points x satisfying $d(x,p)=d(x,q)$ are minimal separating. Here we classify which topological graphs can be realized as minimal separating sets in surfaces of low genus. In general the question of whether a graph can be embedded at all in a surface is a difficult one, so our work is partly computational. We classify graph embeddings which are minimal separating in a given genus and write a computer program to find all such embeddings and underlying graphs.