The content introduces tree-verifiable graph grammars as a syntactic restriction for CMSO-definable graph languages with bounded embeddable tree-width. It discusses regular tree grammars, embeddable tree-width, and the intersection of CMSO-definable and HR-context-free sets of graphs. The paper establishes the completeness of tree-verifiable grammars and their relation to regular graph grammars. It also covers Monadic Second Order Logic with Counting (CMSO) and recognizable sets of graphs. The content includes definitions, theorems, and propositions related to tree-verifiable graph grammars.
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