The monadic second-order logic of graphs XVI : Canonical graph<br> decompositions

TL;DR

This paper demonstrates that various canonical graph decompositions, including split, modular, and Tutte decompositions, are definable in Monadic Second-Order Logic, enabling logical characterization of graph properties.

Contribution

It proves the definability of canonical graph decompositions in Monadic Second-Order Logic, extending previous results and applying to cycle matroids of 2-connected graphs.

Findings

Split decomposition is MSO-definable.

Canonical decompositions can be characterized by MSO formulas.

Graphs with the same cycle matroid as a given graph are MSO-definable.

Abstract

This article establishes that the split decomposition of graphs introduced by Cunnigham, is definable in Monadic Second-Order Logic.This result is actually an instance of a more general result covering canonical graph decompositions like the modular decomposition and the Tutte decomposition of 2-connected graphs into 3-connected components. As an application, we prove that the set of graphs having the same cycle matroid as a given 2-connected graph can be defined from this graph by Monadic Second-Order formulas.