We give $\operatorname{CMSO}$-transductions that, given a graph $G$, output its modular decomposition, its split decomposition and its bi-join decomposition. This improves results by Courcelle [Logical Methods in Computer Science, 2006] who gave such transductions using order-invariant $\operatorname{MSO}$, a strictly more expressive logic than $\operatorname{CMSO}$. Our methods more generally yield $\operatorname{C}_2 \operatorname{MSO}$-transductions that output the canonical decompositions of weakly-partitive set systems and weakly-bipartitive systems of bipartitions.

翻译：我们给出CMSO-传导（transductions），对于给定的图G，输出其模块分解、分裂分解和双连接分解。这改进了Courcelle[逻辑方法在计算机科学，2006]的结果——他使用序不变MSO（一种严格比CMSO更具表达力的逻辑）给出了此类传导。我们的方法更一般地产生了C2MSO-传导，能够输出弱分划集系统的规范分解以及二分划的弱二分系统的规范分解。