On describing trees and quasi-trees from their leaves

2025-03-04

AI Summary

This paper addresses the leaf-node characterization and reconstructibility of O-trees (hierarchically ordered partial orders) and quasi-trees (undirected O-trees). We first establish a first-order axiomatization of the leaf-set relation structure, enabling unique isomorphism identification; then propose a constructive reconstruction of the original structure from its leaves definable in counting monadic second-order logic (CMSO). Our main contributions are: (1) the first purely leaf-based logical characterization of O-trees and quasi-trees; (2) the revelation of deep correspondences between their leaf-relation structures and classical graph decompositions—specifically modular decomposition and rank-width; and (3) a unified, logically definable, and constructively realizable theoretical framework for modular and rank decompositions of countable graphs. The approach bridges order-theoretic structures with graph-theoretic decomposition paradigms through leaf-centric logical specification and reconstruction.

Technology Category

• Knowledge Representation and Reasoning: Description Logics
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Characterize O-trees and quasi-trees from leaf structures.
• Reconstruct O-trees and quasi-trees using CMSO-transductions.
• Apply leaf-based constructions to graph decompositions.

Innovation

• Defines O-trees and quasi-trees using hierarchical orders.
• Characterizes structures via universal first-order sentences.
• Reconstructs trees using CMSO-transductions from leaves.