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Leaky positive semidefinite forcing on graphs

Olivia Elias, Ian Farish, Emrys King, Josh Kyei and Ryan Moruzzi, Jr.

Vol. 18 (2025), No. 4, 719–735
Abstract

We introduce -leaky positive semidefinite forcing and the -leaky positive semidefinite number of a graph, Z+(G), which combines the positive semidefinite color change rule with the addition of leaks to the graph. Furthermore, we determine general properties of Z+(G) and Z+(G) for various graphs, including path graphs, complete graphs, wheel graphs, complete bipartite graphs, trees, hypercubes, and prisms. We also define -leaky positive semidefinite forts with the purpose of unveiling differences between -leaky standard forcing and -leaky positive semidefinite forcing.

Keywords
zero forcing, positive semidefinite forcing, leaky forcing, forts, leaky forts, complete bipartite graphs, trees, Cartesian products
Mathematical Subject Classification
Primary: 05C50, 05C76
Milestones
Received: 15 December 2023
Revised: 6 March 2024
Accepted: 28 March 2024
Published: 30 July 2025

Communicated by Chi-Kwong Li
Authors
Olivia Elias
University of Colorado
Colorado Springs, CO
United States
Ian Farish
California State Polytechnic University
Pomona, CA
United States
Emrys King
Pomona College
Claremont, CA
United States
Josh Kyei
Morehouse College
Atlanta, CA
United States
Ryan Moruzzi, Jr.
Department of Mathematics
California State University, East Bay
Hayward, CA
United States