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Boundary rigidity of systolic and Helly complexes

Martín Blufstein, Jérémie Chalopin, Victor Chepoi

TL;DR

The paper proves that finite systolic, weakly systolic, and Helly complexes are boundary rigid, meaning they are determined up to isomorphism by the combinatorial boundary distances encoded in the boundary distance matrix $D(X)$. It provides constructive reconstruction procedures: for Helly complexes and for 2-dimensional systolic complexes the reconstruction runs in time polynomial in the number of vertices, using a dominated-vertex peeling strategy and updates to the boundary subset and distance matrix. The approach relies on dismantlability and local-to-global properties of the underlying graph classes, employing lemmas tailored to each class (e.g., dominated-vertex results and Helly-type lemmas) to ensure reconstruction. The results advance the discrete boundary rigidity program initiated by Haslegrave et al., offering explicit algorithms, complexity bounds, and several open questions about extending polynomial-time reconstruction to all systolic complexes and about minimal filling properties. Overall, the work bridges combinatorial curvature conditions with rigidity phenomena and provides practical reconstruction methods from boundary data.

Abstract

In this article, we prove that finite (weakly) systolic and Helly complexes can be reconstructed from their boundary distances (computed in their 1-skeleta). Furthermore, Helly complexes and 2-dimensional systolic complexes can be reconstructed by an algorithm that runs in polynomial time with respect to the number of vertices of the complex. Both results can be viewed as a positive contribution to a general question of Haslegrave, Scott, Tamitegama, and Tan (2025). The reconstruction of a finite cell complex from the boundary distances is the discrete analogue of the boundary rigidity problem, which is a classical problem from Riemannian geometry.

Boundary rigidity of systolic and Helly complexes

TL;DR

The paper proves that finite systolic, weakly systolic, and Helly complexes are boundary rigid, meaning they are determined up to isomorphism by the combinatorial boundary distances encoded in the boundary distance matrix . It provides constructive reconstruction procedures: for Helly complexes and for 2-dimensional systolic complexes the reconstruction runs in time polynomial in the number of vertices, using a dominated-vertex peeling strategy and updates to the boundary subset and distance matrix. The approach relies on dismantlability and local-to-global properties of the underlying graph classes, employing lemmas tailored to each class (e.g., dominated-vertex results and Helly-type lemmas) to ensure reconstruction. The results advance the discrete boundary rigidity program initiated by Haslegrave et al., offering explicit algorithms, complexity bounds, and several open questions about extending polynomial-time reconstruction to all systolic complexes and about minimal filling properties. Overall, the work bridges combinatorial curvature conditions with rigidity phenomena and provides practical reconstruction methods from boundary data.

Abstract

In this article, we prove that finite (weakly) systolic and Helly complexes can be reconstructed from their boundary distances (computed in their 1-skeleta). Furthermore, Helly complexes and 2-dimensional systolic complexes can be reconstructed by an algorithm that runs in polynomial time with respect to the number of vertices of the complex. Both results can be viewed as a positive contribution to a general question of Haslegrave, Scott, Tamitegama, and Tan (2025). The reconstruction of a finite cell complex from the boundary distances is the discrete analogue of the boundary rigidity problem, which is a classical problem from Riemannian geometry.

Paper Structure

This paper contains 13 sections, 11 theorems, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graphs and simplicial complexes
  4. Boundary rigidity of simplicial complexes
  5. Bridged and Helly graphs
  6. Boundary rigidity of systolic complexes
  7. Boundary rigidity of Helly complexes
  8. Proof of Theorem \ref{['thm:boundary_rigidity_helly']}
  9. The complexity of the reconstruction
  10. Reconstruction of 2-dimensional systolic complexes
  11. The computation of a dominated vertex
  12. The complexity of the reconstruction
  13. Final remarks

Key Result

Theorem 1.1

Let $X$ be a finite systolic complex. Then $X$ is determined up to isomorphism by the combinatorial distances of vertices of $\partial X$ in $X$. Hence, the class of finite systolic complexes is boundary rigid.

Figures (1)

  • Figure 1: Two simplicial complexes $X_1$ and $X_2$ with dismantlable 1-skeletons. $X_1$ and $X_2$ cannot be distinguished from their boundary information (the boundary of each complex is the downward closure of the thick edges).

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1: Boundary rigidity
  • Lemma 2.2: ChOs
  • Lemma 2.3: BaPe
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 7 more