Journal of Combinatorics

Volume 14 (2023)

Number 4

Generalizations of leaky forcing

Pages: 407 – 424

DOI: https://dx.doi.org/10.4310/JOC.2023.v14.n4.a1

Authors

Joseph S. Alameda (Department of Mathematics, United States Naval Academy, Annapolis, Maryland, U.S.A.)

Jürgen Kritschgau (Department of Mathematics, Carnegie Mellon University, Pittsburgh, Pennsylvania, U.S.A.)

Michael Young (Department of Mathematics, Carnegie Mellon University, Pittsburgh, Pennsylvania, U.S.A.)

Abstract

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.

Keywords

zero forcing, leaky forcing; color-change-rule

2010 Mathematics Subject Classification

Primary 05C57. Secondary 05C15, 05C50.

Received 13 July 2021

Accepted 1 August 2022

Published 14 April 2023