On leaky forcing and resilience

TL;DR

This paper investigates the concept of leaky forcing in graphs, analyzing how resilience to leaks affects zero forcing sets, and provides bounds for various graph families including grids, addressing open questions.

Contribution

It introduces a framework for characterizing $oldsymbol{ extit{ extlangle} extit{ extlangle} extell} $-leaky forcing sets and explores structural implications of $oldsymbol{ extit{ extlangle} extit{ extlangle} extell} $-resilient graphs.

Findings

Robust $( extit{ extlangle} extit{ extlangle} ext{ extlangle})$-leaky forcing sets are also $ extit{ extlangle} extit{ extlangle} ext{ extlangle}$-leaky.

Bounds established for leaky forcing numbers of trees, supertriangles, and grids.

Resolved an open question on the upper bound of 1-leaky forcing number for grid graphs.

Abstract

A leak is a vertex that is not allowed to perform a force during the zero forcing process. Leaky forcing was recently introduced as a new variation of zero forcing in order to analyze how leaks in a network disrupt the zero forcing process. The $\ell$-leaky forcing number of a graph is the size of the smallest zero forcing set that can force a graph despite $\ell$ leaks. A graph $G$ is $\ell$-resilient if its zero forcing number is the same as its $\ell$-leaky forcing number. In this paper, we analyze $\ell$-leaky forcing and show that if an $(\ell-1)$-leaky forcing set $B$ is robust enough, then $B$ is an $\ell$-leaky forcing set. This provides the framework for characterizing $\ell$-leaky forcing sets. Furthermore, we consider structural implications of $\ell$-resilient graphs. We apply these results to bound the $\ell$-leaky forcing number of several graph families including trees, supertriangles, and grid graphs. In particular, we resolve a question posed by Dillman and Kenter concerning the upper bound on the $1$-leaky forcing number of grid graphs.