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.

翻译：我们考虑弦图的一个自然推广，其中每个最小分离子集导出一个独立数至多为 $2$ 的子图。这类图等价地可定义为不包含完全二分图 $K_{2,3}$ 作为导出子式的图，即无法通过一系列边收缩和顶点删除操作从该图中得到 $K_{2,3}$ 的图。我们开发了一种多项式时间算法来识别这些图。该算法依赖于 $K_{2,3}$ 导出子式自由图的一个刻画，即排除特定的导出子图（称为 Truemper 构型）。