Finding minimal forbidden minors using a finite congruence.

*(English)*Zbl 0764.68122
Automata, languages and programming, Proc. 18th Int. Colloq., Madrid/Spain 1991, Lect. Notes Comput. Sci. 510, 532-543 (1991).

Summary: [For the entire collection see Zbl 0753.00027.]

We give an effective way to compute the minimal forbidden minors for a minor-closed class of graphs of bounded tree-width from an algorithm that decides a finite congruence that recognizes the class. We prove constructively that every minor closed class of graphs of bounded tree- width that is recognized by a finite congruence has a finite number of minimal forbidden minors. Our proof gives a bound of the size of a minimal forbidden minor. We define explicitly a relation \(\sim\), prove that it is a finite congruence that recognizes the graphs of tree-width at most \(w\), and show how to decide it. Hence, we can find the minimal forbidden minors for graphs of tree-width at most \(w\) and bounds on their sizes. An algorithm that recognizes graphs of tree-width at most \(w\) in linear time is also obtained.

We give an effective way to compute the minimal forbidden minors for a minor-closed class of graphs of bounded tree-width from an algorithm that decides a finite congruence that recognizes the class. We prove constructively that every minor closed class of graphs of bounded tree- width that is recognized by a finite congruence has a finite number of minimal forbidden minors. Our proof gives a bound of the size of a minimal forbidden minor. We define explicitly a relation \(\sim\), prove that it is a finite congruence that recognizes the graphs of tree-width at most \(w\), and show how to decide it. Hence, we can find the minimal forbidden minors for graphs of tree-width at most \(w\) and bounds on their sizes. An algorithm that recognizes graphs of tree-width at most \(w\) in linear time is also obtained.

##### MSC:

68R10 | Graph theory (including graph drawing) in computer science |

68T10 | Pattern recognition, speech recognition |

05C35 | Extremal problems in graph theory |

05C05 | Trees |

05C38 | Paths and cycles |