## Abstract

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.

Original language | English (US) |
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Pages (from-to) | 248-280 |

Number of pages | 33 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 152 |

DOIs | |

State | Published - Jan 2022 |

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

## Keywords

- Induced subgraph
- Minimal separator
- Prism
- Pyramid
- Theta
- Turtle