## Tolerance GraphsThe study of algorithmic graph theory and structured families of graphs is an important branch of discrete mathematics. It finds numerous applications, from data transmission through networks to efficiently scheduling aircraft and crews, as well as contributing to breakthroughs in genetic analysis and studies of the brain. Especially important have been the theory and applications of new intersection graph models such as generalizations of permutation graphs and interval graphs. One of these is the study of tolerance graphs and tolerance orders. This book contains the first thorough study of tolerance graphs and related topics, indeed the authors have included proofs of major results previously unpublished in book form. It will act as a springboard for researchers, and especially graduate students, to pursue new directions of investigation. With many examples and exercises it is also suitable for use as the text for a graduate course in graph theory. |

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### Contents

definitions and examples | 5 |

Ordered sets | 13 |

Early work on tolerance graphs | 29 |

Trees cotrees and bipartite graphs | 53 |

Interval probe graphs and sandwich problems | 63 |

Bitolerance and the ordered sets perspective | 84 |

Unit and 50 tolerance orders | 98 |

Comparability invariance results | 109 |

Recognition of bounded bitolerance orders | 124 |

Algorithms on tolerance graphs | 135 |

The hierarchy of classes of bounded bitolerance orders | 146 |

Tolerance models of paths and subtrees of a tree | 164 |

Directed tolerance graphs | 219 |

Open questions and further directions of research | 249 |

260 | |

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### Common terms and phrases

0-tolerance algorithm assigned autonomous set bipartite Bogart bounded bitolerance digraph bounded bitolerance orders bounded bitolerance representation bounded tolerance graph chain graph Chapter chordal graphs chordless cycle classes of bounded classes of unit clique cocomparability coloring comparability graph contradiction define definition digraph G elementary reversal elements Exercise Golumbic graph G hierarchy incomparability induced subgraph intersection graph interval graph interval orders interval probe graph interval representation Lemma Let G linear McMorris Monma NeST graphs NeST representation non-probes NP-complete ordered set parallelogram partition perfect graphs permutation graph point-core bitolerance representation polynomial proof of Theorem proper tolerance Proposition prove Remark representation of G result separating examples shown in Figure split graphs splitting points stable set strongly chordal sum-tolerance Suppose Theorem threshold graph tolerant points totally bounded transitive orientation trapezoid graph trapezoid representation Trenk Trotter unit tolerance graphs unit tolerance order unit tolerance representation v e V(G vertices