CMSO-transducing tree-like graph decompositions

TL;DR

This paper develops CMSO-transductions to compute various graph decompositions, improving upon prior results that used more expressive logic, and extends to canonical decompositions of set systems.

Contribution

It introduces CMSO-transductions for graph decompositions, surpassing previous methods based on order-invariant MSO logic, and generalizes to set system decompositions.

Findings

CMSO-transductions for modular, split, and bi-join decompositions

Improves upon Courcelle's earlier results using more expressive logic

Extends to canonical decompositions of set systems

Abstract

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.