On the Connectivity of the Vietoris-Rips Complex of a Hypercube Graph

Martin Bendersky; Jelena Grbic

paper

On the Connectivity of the Vietoris-Rips Complex of a Hypercube Graph

arXiv:2311.06407

Abstract

We bring in the techniques of independence complexes and the notion of total dominating sets of a graph to bear on the question of the connectivity of the Vietoris-Rips complexes $V R (Q_{n}; r)$ of an $n$ -hypercube graph. We obtain a lower bound for the connectivity of $V R (Q_{n}; r)$ for an arbitrary $n$ -dimension hypercube and at all scale parameters $r$ . The obtained bounds disprove the conjecture of Shukla that $\VR$ is $r$ -connected.