Leaky Forcing: Extending Zero Forcing Results to a Fault-Tolerant Setting

TL;DR

This paper introduces leaky forcing, a fault-tolerant extension of zero forcing, providing exact and bounded results for various graph classes and analyzing the impact of vertex and edge removals.

Contribution

It extends zero forcing results to leaky forcing, offering complete solutions for unicyclic graphs, bounds for Petersen graphs, and characterizations of graphs with extreme leaky forcing numbers.

Findings

Leaky forcing numbers are determined for all unicyclic graphs.

Upper bounds are established for generalized Petersen graphs.

Bounds are provided for the effects of vertex and edge removal on leaky forcing.

Abstract

We study a recent variation of zero forcing called leaky forcing. Zero forcing is a propagation process on a network whereby some nodes are initially blue with all others white. Blue vertices can "force" a white neighbor to become blue if all other neighbors are blue. The goal is to find the minimum number of initially blue vertices to eventually force all vertices blue after exhaustively applying the forcing rule above. Leaky forcing is a fault-tolerant variation of zero forcing where certain vertices (not necessarily initially blue) cannot force. The goal in this context is to find the minimum number of initially blue vertices needed that can eventually force all vertices to be blue, regardless of which small number of vertices can't force. This work extends results from zero forcing in terms of leaky forcing. In particular, we provide a complete determination of leaky forcing numbers for all unicyclic graphs and upper bounds for generalized Petersen graphs. We also provide bounds for the effect of both edge removal and vertex removal on the $\ell$-leaky forcing number. Finally, we completely characterize connected graphs that have the minimum and maximum possible $1$-leaky forcing number (i.e., when $Z_{1}(G) = 2$ and when $Z_{1}(G) = |V(G)|-1$).