Regularity Equals Monadic Second-Order Definability for Quasi-trees

Abstract

Quasi-trees generalize trees in that the unique “path” between two nodes may be infinite and have any finite or countable order type, in particular that of rational numbers. They are used to define the rank-width of a countable graph in such a way that it is the least upper-bound of the rank-widths of its finite induced subgraphs. $Join-trees$ are the corresponding directed “trees” and they are also useful to define the modular decomposition of a countable graph. We define algebras with finitely many operations that generate (via infinite terms) these generalized trees. We prove that the associated regular objects (those defined by regular terms) are exactly the ones definable by (i.e., are the unique models of) monadic second-order sentences. These results use and generalize a similar result by W. Thomas for countable linear orders.