In this paper we introduce and investigate a new class of graphs called algebraic forests for which isomorphism testing can be done in time . The class of algebraic forests admits a membership test of the same complexity, it includes cographs, trees and interval graphs, and even a joint superclass of the latter two, namely, rooted-directed path graphs. In fact, our class is much larger than these classes, since every graph is an induced subgraph of some algebraic forest. The key point of our approach is the study of the class of forestal cellular algebras defined inductively from one-point algebras by taking direct sums and wreath products. In fact, algebraic forests are exactly the graphs the cellular algebras of which are forestal. We prove that each weak isomorphism of two forestal algebras is induced by a strong isomorphism. This implies that all forestal algebras are compact cellular algebras and so all algebraic forests are weakly compact graphs. We also present a complete description of cellular algebras of disconnected graphs.
