Leaky positive semidefinite forcing on graphs

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

doi:10.2140/involve.2025.18.719

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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

DOI: 10.2140/involve.2025.18.719

Abstract

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

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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

Vol. 18, No. 4, 2025