Journal of Combinatorics
Volume 14 (2023)
Generalizations of leaky forcing
Pages: 407 – 424
Motivated by the inverse eigenvalue problem, 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.
zero forcing, leaky forcing; color-change-rule
2010 Mathematics Subject Classification
Primary 05C57. Secondary 05C15, 05C50.
The second-named author’s research is supported by NSF Grant DMS-1839918.
The third-named author’s research is supported by NSF Grant DMS-1719841.
Received 13 July 2021
Accepted 1 August 2022
Published 14 April 2023