Finding minimal forbidden minors using a finite congruence. (English) Zbl 0764.68122

Lagergren, Jens; Arnborg, Stefan

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

Keywords:

linear time algorithm; minimal forbidden minors; graphs of bounded tree- width; finite congruence; bound

Citations:

Zbl 0753.00027