A lower bound on the failed zero-forcing number of a graph

Swanson, Nicolas; Ufferman, Eric

doi:10.2140/involve.2023.16.493

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A lower bound on the failed zero-forcing number of a graph

Nicolas Swanson and Eric Ufferman

Vol. 16 (2023), No. 3, 493–504

DOI: 10.2140/involve.2023.16.493

Abstract

Given a finite simple graph $G = (V, E)$ and a set of vertices marked as filled, we consider a color-change rule known as zero forcing. A set $S$ is a zero-forcing set if filling $S$ and applying all possible instances of the color-change rule causes all vertices in $V$ to be filled. A failed zero-forcing set is a set of vertices that is not a zero-forcing set. Given a graph $G$ , the failed zero-forcing number $F (G)$ is the maximum order of a failed zero-forcing set. Fetcie, Jacob and Saavedra (Involve 8:1 (2015), 99–117) asked whether given any $k$ there exists an $ℓ$ such that all graphs with at least $ℓ$ vertices must satisfy $F (G) \geq k$ . We answer this question affirmatively by proving that for a graph $G$ with $n$ vertices, $F (G) \geq ⌊ (n - 1) ∕ 2 ⌋$ .

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Keywords

failed zero forcing

Mathematical Subject Classification

Primary: 05C99

Milestones

Received: 14 April 2022

Revised: 2 August 2022

Accepted: 5 August 2022

Published: 10 August 2023

Communicated by Kenneth S. Berenhaut

Authors

Nicolas Swanson
Department of Mathematics Virginia Tech University Blacksburg, VA United States

Eric Ufferman
Department of Mathematics Virginia Tech University Blacksburg, VA United States

Vol. 16, No. 3, 2023