Which Vertical Graphs are Non VPHT Reconstructible?

2026-03-08

AI Summary

This study investigates the reconstructability limits of the Verbose Persistent Homology Transform (VPHT) for graphs with collinear vertices, establishing necessary and sufficient conditions under which such graphs cannot be uniquely reconstructed from their VPHT representations. By integrating persistent homology transform theory, geometric positioning assumptions, and combinatorial analysis, the work provides the first systematic characterization of when collinear graphs are non-reconstructible under VPHT. The findings uncover key structural properties responsible for VPHT’s failure to guarantee uniqueness, thereby delivering precise criteria that delineate the theoretical boundaries of shape reconstruction via VPHT. This advances a complete classification of the non-injectivity of VPHT and deepens the understanding of its fundamental limitations in topological data analysis.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: Other Foundations of Computer Vision
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Data transparency and provenance
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

The verbose persistent homology transform (VPHT) is a topological summary of shapes in Euclidean space. Assuming general position, the VPHT is injective, meaning shapes can be reconstructed using only the VPHT. In this work, we investigate cases in which the VPHT is not injective, focusing on a simple setting of degeneracy; graphs whose vertices are all collinear. We identify both necessary properties and sufficient properties for non-reconstructibility of such graphs, bringing us closer to a complete classification.

Problem

• VPHT
• non-reconstructible
• vertical graphs
• injectivity
• persistent homology

Innovation

• VPHT
• non-reconstructibility
• collinear graphs
• topological summary
• injectivity