Recognizable sets of graphs: equivalent definitions and closure properties

The notion of a recognizable set of words, trees or graphs is relative to an algebraic structure on the set of words, trees or graphs respectively. We establish that several algebraic structures yield the same notion of a recognizable set of graphs. This notion is equivalent to that of a fully cutset-regular set of graphs introduced by Fellows and Abrahamson. We also establish that the class of recognizable sets of graphs is closed under the operations considered in these various equivalent definitions. This fact is not a standard consequence of the definition of recognizability.