  Minimal separators in graphs are an important concept in algorithmic graph
theory. In particular, many problems that are NP-hard for general graphs are
known to become polynomial-time solvable for classes of graphs with a
polynomially bounded number of minimal separators. Several well-known graph
classes have this property, including chordal graphs, permutation graphs,
circular-arc graphs, and circle graphs. We perform a systematic study of the
question which classes of graphs defined by small forbidden induced subgraphs
have a polynomially bounded number of minimal separators. We focus on sets of
forbidden induced subgraphs with at most four vertices and obtain an almost
complete dichotomy, leaving open only two cases.
