The Zero Forcing Number of Twisted Hypercubes

TL;DR

This paper investigates the zero forcing number of twisted hypercubes, a generalization of hypercubes, revealing that their zero forcing sets can be smaller than those of standard hypercubes, which has implications for graph infection processes.

Contribution

It introduces a new construction of zero forcing sets for twisted hypercubes, showing they can be smaller than in hypercubes, advancing understanding of graph infection dynamics.

Findings

Constructed twisted hypercubes with smaller zero forcing sets

Demonstrated zero forcing sets of size below hypercube minimums

Provided a new perspective on graph infection processes

Abstract

Twisted hypercubes are graphs that generalize the structure of the hypercube by relaxing the symmetry constraint while maintaining degree-regularity and connectivity. We study the zero forcing number of twisted hypercubes. Zero forcing is a graph infection process in which a particular colour change rule is iteratively applied to the graph and an initial set of vertices. We use the alternative framing of forcing arc sets to construct a family of twisted hypercubes of dimension k$\geq 3$ with zero forcing sets of size $2^{k-1}-2^{k-3}+1$, which is below the minimum zero forcing number of the hypercube.