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.

翻译：冗余持久同调变换（VPHT）是欧几里得空间中形状的一种拓扑摘要。在一般位置假设下，VPHT是单射的，这意味着仅使用VPHT即可重构形状。本文研究VPHT非单射的情形，重点关注一类简单的退化情况：所有顶点共线的图。我们为此类图的不可重构性确定了必要条件和充分条件，从而向完整分类更近一步。