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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.

##### 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