Detecting \(K_{2,3}\) as an Induced Minor

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.