We give a characterization of those sets of graphs that are both definable in Counting Monadic Second Order Logic (CMS) and context-free, i.e., least solutions of Hyperedge-Replacement (HR)-grammars introduced by Courcelle and Engelfriet. We give the following equivalent characterizations: (a) a set of graphs is recognizable (in the algebra that consists of all graphs and HR-operations) and has bounded tree-width; further, we refine this condition and show equivalence with recognizability in a finite-sort subalgebra of the graph algebra; (b) the set is parsable, i.e., there is an MS-definable transduction from graphs to a set of derivation trees labelled by HR-operations, such that the set of graphs is the image of this set of trees under the evaluation of the HR-operations; (c) the set of graphs is the image of unranked recognizable set of trees under an MS-definable transduction whose inverse is also MS-definable. The main goal of this paper is to present the above characterization, of which several directions are already known, in an accessible and unified way. We rely on a novel connection between two seminal results, a logical characterization of context-free graph languages in terms of tree to graph MS-definable transductions, by Courcelle and Engelfriet~, and a proof that an optimal-width tree decomposition of a graph can be built by an MS-definable transduction, by Bojanczyk and Pilipczuk.

翻译：本文给出了那些在计数单子二阶逻辑（CMS）中可定义且是上下文无关图集（即由Courcelle和Engelfriet引入的超边替换（HR）文法的最小解）的特征刻画。我们提出了如下等价刻画：(a) 图集是可识别的（在包含所有图及HR操作的代数中）且具有有界树宽；进一步，我们细化该条件，证明了其等价于在该图代数的一个有限类子代数中的可识别性；(b) 图集是可解析的，即存在一个从图到由HR操作标记的派生树集的MS可定义转导，使得该图集是该树集在HR操作求值下的像；(c) 图集是一个无排秩可识别树集在一个其逆也是MS可定义的MS可定义转导下的像。本文的主要目标是，以易于理解且统一的方式呈现上述特征刻画（其中部分方向已有已知结果）。我们依赖于两个开创性结果之间的新联系：一是Courcelle与Engelfriet关于用树到图的MS可定义转导来逻辑刻画上下文无关图语言；二是Bojanczyk与Pilipczuk关于可通过MS可定义转导构建图的最优宽度树分解的证明。