Which Vertical Graphs are Non VPHT Reconstructible?
This work addresses a theoretical problem for mathematicians and computer scientists working with topological data analysis, specifically concerning the limitations of the VPHT for shape reconstruction.
This paper investigates the conditions under which the verbose persistent homology transform (VPHT) fails to be injective for graphs with collinear vertices. The authors identify both necessary and sufficient properties for non-reconstructibility in this specific degenerate setting.
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.