Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Which Vertical Graphs are Non VPHT Reconstructible?

Jette Gutzeit, Kalani Kistler, Tim Ophelders, Anna Schenfisch

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.

Which Vertical Graphs are Non VPHT Reconstructible?

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.
Paper Structure (16 sections, 6 theorems, 6 figures, 2 algorithms)

This paper contains 16 sections, 6 theorems, 6 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Theoretical Results for Vertical Graphs
  4. Computational Approach
  5. Computational Limits
  6. Application Feature Overview
  7. Numerical Results
  8. Discussion and Conclusion
  9. Computational Approach
  10. Additional features of the GUI
  11. Algorithms
  12. Computing the VPHT
  13. Computing Colliding Graphs
  14. Computing Collision Sets
  15. Searching for Alternating Cycles and Finding Minimal Partitions
  16. ...and 1 more sections

Key Result

Theorem 9

Consider a type $\mathcal{G}$ graph, such that each vertex has no more than two lower neighbors and no more than two upper neighbors. Then, for any choice of alternating cycles $C^1,\ldots,C^k$ partitioning $G$, the resulting graphs $\mathcal{G}_1$ and $\mathcal{G}_2$ are non VPHT-reconstructible.

Figures (6)

  • Figure 1: The lower-star filtration of this graph with respect to the vertical up direction $s$ has a step for each vertex height, adding; (1) the bottom vertex (which creates a connected component), (2) the middle vertex and first edge (no effect on homology), and (3) the top vertex and remaining edges (creating a cycle). A corresponding index filtration is shown in the boxes, adding vertices and edges individually. This causes instantaneous births/deaths of components, which appear as on-diagonal points in the verbose diagram (right). The circle denotes the cycle born at height (3) and the black dots denote connected components.
  • Figure 2: The above Venn diagram shows the classification of different graphs that will be discussed in the following pages. All inclusions are strict, where \ref{['lem:anothercycle']} distinguishes the pink class from the red and \ref{['fig:multiple edges']} distinguishes the purple class from the pink.
  • Figure 3: The disjoint union (left) of a simple colliding pair (middle and right). We observe the alternating cycle if we consider all red edges to be oriented up (without loss of generality) and all blue edges to be oriented down.
  • Figure 4: Verbose diagram for the red and blue graphs of \ref{['fig:firstEx']} when filtering in the up direction.
  • Figure 5: On the left $\mathcal{G}$, made up of two vertical graphs with alternating cycles; one includes all red and blue edges, the other the cyan and magenta edge. On the right the corresponding $\mathcal{G}_1$ and $\mathcal{G}_2$.
  • ...and 1 more figures

Theorems & Definitions (14)

  • Definition 2: Verbose Persistent Homology Transform
  • Definition 3: VPHT-Reconstructible
  • Definition 4: Vertical Graph
  • Definition 6: Simple Colliding Pair and Alternating Cycle
  • Definition 7
  • Definition 8
  • Theorem 9: Special Colliding Pairs are Non Reconstructible
  • Lemma 9: Type $\mathcal{G}$ Equivalence
  • Theorem 10: Non-Reconstructible implies Colliding Pair
  • Lemma 10
  • ...and 4 more