Finding minimal forbidden minors using a finite congruence

Abstract

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 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.