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

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.