Equational axiomatization of graph operations

The source page states the original problem together with its recorded qualifications and progress updates as follows.

Given a set of graph operations like disjoint union, edge-complement, quantifier-free definable operations (like relabellings). When can one axiomatize its equational theory by finitely many equational rules or by infinitely many rules describable in finitary ways?

See [Courcelle, 2005].