  Hyperedge-Replacement grammars (HR) have been introduced by Courcelle in
order to extend the notion of context-free sets from words and trees to graphs
of bounded tree-width. While for words and trees the syntactic restrictions
that guarantee that the associated languages of words resp. trees are regular -
and hence, MSO-definable - are known, the situation is far more complicated for
graphs. Here, Courcelle proposed the notion of regular graph grammars, a
syntactic restriction of HR grammars that guarantees the definability of the
associated languages of graphs in Counting Monadic Second Order Logic (CMSO).
However, these grammars are not complete in the sense that not every
CMSO-definable set of graphs of bounded tree-width can be generated by a
regular graph grammar. In this paper, we introduce a new syntactic restriction
of HR grammars, called tree-verifiable graph grammars, and a new notion of
bounded tree-width, called embeddable bounded tree-width, where the later
restricts the trees of a tree-decomposition to be a subgraph of the analyzed
graph. The main property of tree-verifiable graph grammars is that their
associated languages are CMSO-definable and that the have bounded embeddable
tree-width. We show further that they strictly generalize the regular graph
grammars of Courcelle. Finally, we establish a completeness result, showing
that every language of graphs that is CMSO-definable and of bounded embeddable
tree-width can be generated by a tree-verifiable graph grammar.
