Boundary rigidity of systolic and Helly complexes

TL;DR

This paper demonstrates that finite systolic and Helly complexes can be reconstructed from boundary distances, with polynomial-time algorithms for certain cases, advancing the understanding of boundary rigidity in discrete geometric structures.

Contribution

It introduces polynomial-time algorithms for reconstructing Helly and 2D systolic complexes from boundary data, addressing a discrete analogue of a classical geometric problem.

Findings

Finite (weakly) systolic and Helly complexes are reconstructible from boundary distances.

Helly and 2D systolic complexes can be reconstructed in polynomial time.

Results contribute to the boundary rigidity problem in discrete geometry.

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.