Which Vertical Graphs are Non VPHT Reconstructible?

TL;DR

This paper investigates the conditions under which certain vertical graphs, specifically collinear vertex graphs, cannot be reconstructed from their verbose persistent homology transform (VPHT), advancing understanding of topological shape summaries.

Contribution

It identifies necessary and sufficient conditions for non-reconstructibility of collinear vertex graphs from VPHT, extending the classification of shapes recoverable by topological summaries.

Findings

VPHT is injective for general position shapes

Collinear vertex graphs can be non-reconstructible from VPHT

Provides criteria for non-reconstructibility of specific graph classes

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.