On describing trees and quasi-trees from their leaves
Bruno Courcelle
TL;DR
The paper broadens the theory of trees by introducing O-trees and quasi-trees and developing leaf-based, logic-driven descriptions that characterize these structures up to isomorphism. It develops universal FO axioms for leaf structures of join-trees and extends to infinite cases via CMSO-constructible classes, enabling reconstruction of join-trees from leaves through MSO/CMSO transductions. It connects these constructions to canonical graph decompositions such as modular decomposition and rank-width by constructing layouts from leaf data, and extends the framework to leafy quasi-trees, partial quasi-trees, and their topological embeddings. Overall, the work integrates logical definability with combinatorial graph decompositions for countable graphs, providing a unified approach to describe, reconstruct, and utilize tree-like structures in graph theory.
Abstract
Generalized trees, we call them O-trees, are defined as hierarchical partial orders, i.e., such that the elements larger than any one are linearly ordered. Quasi-trees are, roughly speaking, undirected O-trees. For O-trees and quasi-trees, we define relational structures on their leaves that characterize them up to isomorphism. These structures have characterizations by universal first-order sentences. Furthermore, we consider cases where O-trees and quasi-trees can be reconstructed from their leaves by CMSO-transductions. These transductions are transformations of relational structures defined by monadic second-order (MSO) formulas. The letter "C" for counting refers to the use of set predicates that count cardinalities of finite sets modulo fixed integers. O-trees and quasi-trees make it possible to define respectively, the modular decomposition and the rank-width of a countable graph. Their constructions from their leaves by transductions of different types apply to rank-decompositions, and to modular decomposition and to other canonical graph decompositions.