The monadic second-order logic of graphs XI: Hierarchical decompositions of connected graphs

Abstract

We prove that the unique decomposition of connected graphs defined by Tutte is definable by formulas of monadic second-order logic. This decomposition has two levels: every connected graph is a tree of “2-connected components” called blocks; every 2-connected graph is a tree of so-called 3- blocks. Our proof uses 2- dags which are certain acyclic orientations of the considered graphs. We also obtain a unique decomposition theorem for 2-dags and a definability of this decomposition in monadic second-order logic.