  We consider a natural generalization of chordal graphs, in which every
minimal separator induces a subgraph with independence number at most $2$. Such
graphs can be equivalently defined as graphs that do not contain the complete
bipartite graph $K_{2,3}$ as an induced minor, that is, graphs from which
$K_{2,3}$ cannot be obtained by a sequence of edge contractions and vertex
deletions.
  We develop a polynomial-time algorithm for recognizing these graphs. Our
algorithm relies on a characterization of $K_{2,3}$-induced minor-free graphs
in terms of excluding particular induced subgraphs, called Truemper
configurations.
