Upper bound for the $(d-2)$-leaky forcing number of $Q_d$ and $\ell$-leaky forcing number of $GP(n,1)$

TL;DR

This paper establishes an upper bound for the $(d-2)$-leaky forcing number of hypercubes and explores the relationship between leaky forcing sets and zero-forcing sets in graphs.

Contribution

It extends previous results by providing an upper bound for the leaky forcing number of hypercubes and investigates the connection between leaky and zero-forcing sets.

Findings

The $(d-2)$-leaky forcing number of $Q_d$ is at most $2^{d-1}$.

Analysis of the relationship between minimum $ ext{leaky}$-forcing and zero-forcing sets.

Extension of leaky forcing concepts to specific graph families.

Abstract

Leaky-forcing is a recently introduced variant of zero-forcing that has been studied for families of graphs including paths, cycles, wheels, grids, and trees. In this paper, we extend previous results on the leaky forcing number of the d-dimensional hypercube, $Q_d$, to show that the $(d-2)$-leaky forcing number of $Q_d$ is at most $2^{d-1}$. We also examine a question about the relationship between the size of a minimum $\ell$-leaky-forcing set and a minimum zero-forcing set for a graph $G$.