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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

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 such that all graphs with at least vertices must satisfy F(G) k. We answer this question affirmatively by proving that for a graph G with n vertices, F(G) (n 1)2.

failed zero forcing
Mathematical Subject Classification
Primary: 05C99
Received: 14 April 2022
Revised: 2 August 2022
Accepted: 5 August 2022
Published: 10 August 2023

Communicated by Kenneth S. Berenhaut
Nicolas Swanson
Department of Mathematics
Virginia Tech University
Blacksburg, VA
United States
Eric Ufferman
Department of Mathematics
Virginia Tech University
Blacksburg, VA
United States