# A technique for recognizing graphs of bounded treewidth with application to subclasses of partial 2-paths

## Abstract

Regarding members of a class of graphs as values of algebraic expressions allows definition of a congruence such that the given class is the union of some of the equivalence classes. In many cases this congruence has a finite number of equivalence classes. Such a congruence can be used to generate a reduction system that decides the class membership in linear time. However, a congruence for a given problem is often difficult to determine.

We describe a technique that produces an algebra and a congruence relation on its carrier for some classes of graphs. Our technique builds on considering possible representations of the generated graphs as graphs of the desired class. By introduction of a labeling describing the “most parsimonious” such representations, we can work with small labeled graphs instead of large unlabeled ones, and some of the case analysis can be delegated to the algebraic machinery used. The congruence relation is subsequently used to construct a labeled graph reduction system based on a reduction system recognizing a larger class than the one sought.

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