Generalizations of Leaky Forcing

TL;DR

This paper explores various generalizations of leaky forcing in graphs, analyzing how vertex and edge leaks affect the zero forcing process and establishing their equivalence.

Contribution

It introduces and proves the equivalence of different leaky forcing variants, including vertex, edge, and specified leaks, and their mixed forms.

Findings

$oldsymbol{ ext{Leaky forcing variants are equivalent}}$

Mixed leaks do not change the forcing number

Analysis extends zero forcing to more realistic leak scenarios

Abstract

Vertex leaky forcing was recently introduced as a new variation of zero forcing in order to show how vertex leaks can disrupt the zero forcing process in a graph. An edge leak is an edge that is not allowed to be forced across during the zero forcing process. The $\ell$-edge-leaky forcing number of a graph is the size of a smallest zero forcing set that can force the graph blue despite $\ell$ edge leaks. This paper contains an analysis of the effect of edge leaks on the zero forcing process instead of vertex leaks. Furthermore, specified $\ell$-leaky forcing is introduced. The main result is that $\ell$-leaky forcing, $\ell$-edge-leaky forcing, and specified $\ell$-leaky forcing are equivalent. Furthermore, all of these different kinds of leaks can be mixed so that vertex leaks, edge leaks, and specified leaks are used. This mixed $\ell$-leaky forcing number is also the same as the (vertex) $\ell$-leaky forcing number.