Year: 2025
Type: preprint
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. This partial order generalizes the ancestor relation of a rooted tree. As for rooted trees, the root of an O-tree is its unique maximal element if there is such; however, an O-tree may have no root. 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 axiomatizations by universal first-order sentences. Our main purpose is to reconstruct O-trees and quasi-trees from their leaves and relevant relations on leaves, by logically defined transformations, called transductions. Monadic second-order (MSO) transductions are defined by monadic second-order (MSO) formulas. We use actually CMSO transductions where the letter “C”, meaning 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, to modular decomposition and to other canonical graph decompositions.
Source: arXiv (Cornell University)
Author: Courcelle, Bruno
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4415335305
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