### Abstract

A vertex v in a graph G is said to be avoidable if every induced two-edge path with midpoint v is contained in an induced cycle. Generalizing Dirac’s theorem on the existence of simplicial vertices in chordal graphs, Ohtsuki et al. proved in 1976 that every graph has an avoidable vertex. In a different generalization, Chvátal et al. gave in 2002 a characterization of graphs without long induced cycles based on the concept of simplicial paths. We introduce the concept of avoidable induced paths as a common generalization of avoidable vertices and simplicial paths. We propose a conjecture that would unify the results of Ohtsuki et al. and of Chvátal et al. The conjecture states that every graph that has an induced k-vertex path also has an avoidable k-vertex path. We prove that every graph with an edge has an avoidable edge, thus establishing the case k= 2 of the conjecture. Furthermore, we point out a close relationship between avoidable vertices in a graph and its minimal triangulations and identify new algorithmic uses of avoidable vertices. More specifically, applying Lexicographic Breadth First Search and bisimplicial elimination orderings, we derive a polynomial-time algorithm for the maximum weight clique problem in a class of graphs generalizing the class of 1-perfectly orientable graphs and its subclasses chordal graphs and circular-arc graphs.

Original language | English (US) |
---|---|

Title of host publication | Algorithms and Data Structures - 16th International Symposium, WADS 2019, Proceedings |

Editors | Zachary Friggstad, Mohammad R. Salavatipour, Jörg-Rüdiger Sack |

Publisher | Springer Verlag |

Pages | 126-139 |

Number of pages | 14 |

ISBN (Print) | 9783030247652 |

DOIs | |

State | Published - Jan 1 2019 |

Event | 16th International Symposium on Algorithms and Data Structures, WADS 2019 - Edmonton, Canada Duration: Aug 5 2019 → Aug 7 2019 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
---|---|

Volume | 11646 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 16th International Symposium on Algorithms and Data Structures, WADS 2019 |
---|---|

Country | Canada |

City | Edmonton |

Period | 8/5/19 → 8/7/19 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

## Fingerprint Dive into the research topics of 'Avoidable vertices and edges in graphs'. Together they form a unique fingerprint.

## Cite this

*Algorithms and Data Structures - 16th International Symposium, WADS 2019, Proceedings*(pp. 126-139). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11646 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-030-24766-9_10