We give a characterization of the sets of graphs that are both definable in Counting Monadic Second Order Logic (CMSO) and context-free, i.e., least solutions of Hyperedge-Replacement (HR) grammars introduced by Courcelle and Engelfriet. We prove the equivalence of these sets with: (a) recognizable sets (in the algebra of graphs with HR-operations) of bounded tree-width; we refine this condition further and show equivalence with recognizability in a finitely generated subalgebra of the HR-algebra of graphs; (b) parsable sets, for which there is an MSO-definable transduction from graphs to a set of derivation trees labelled by HR operations, such that the set of graphs is the image of the set of derivation trees under the canonical evaluation of the HR operations; (c) images of recognizable unranked sets of trees under an MSO-definable transduction, whose inverse is also MSO-definable. We rely on a novel connection between two seminal results, a logical characterization of context-free graph languages in terms of tree to graph MSO-definable transductions, by Courcelle and Engelfriet and a proof that an optimal-width tree decomposition of a graph can be built by an MSO-definable transduction, by Bojanczyk and Pilipczuk.

翻译：我们给出了同时可在计数单子二阶逻辑（CMSO）中定义且具有上下文无关性（即Courcelle和Engelfriet提出的超边替换文法的最小解）的图集的特征描述。我们证明这些图集与以下概念等价：(a) 在HR操作图代数中具有有界树宽的可识别集；我们进一步细化该条件，证明其等价于在HR图代数的有限生成子代数中的可识别性；(b) 可解析集，即存在从图到由HR操作标记的派生树集的MSO可定义转移，使得图集恰好是该派生树集在HR操作规范评价下的像；(c) 在MSO可定义转移（其逆转移也是MSO可定义）下，可识别的无秩树集的像。我们的证明基于两个开创性结果的新关联：Courcelle与Engelfriet关于上下文无关图语言在树到图MSO可定义转移下的逻辑特征描述，以及Bojanczyk与Pilipczuk关于图的最优宽度树分解可通过MSO可定义转移构造的证明。