We show that graphs that do not contain a theta, pyramid, prism, or turtle as an induced subgraph have polynomially many minimal separators. This result is the best possible in the sense that there are graphs with exponentially many minimal separators if only three of the four induced subgraphs are excluded. As a consequence, there is a polynomial time algorithm to solve the maximum weight independent set problem for the class of (theta, pyramid, prism, turtle)-free graphs. Since every prism, theta, and turtle contains an even hole, this also implies a polynomial time algorithm to solve the maximum weight independent set problem for the class of (pyramid, even hole)-free graphs.

我们表明，不包含 theta、金字塔、棱柱或海龟作为诱导子图的图具有多项式许多最小分隔符。如果仅排除四个诱导子图中的三个，则该结果是最好的，因为存在具有指数级多个最小分隔符的图。因此，存在多项式时间算法来解决（θ、金字塔、棱柱、乌龟）无图类的最大权重独立集问题。由于每个棱镜、θ 和海龟都包含一个偶数孔，这也意味着使用多项式时间算法来解决无（金字塔，偶数孔）图类的最大权重独立集问题。