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Article

All Graphs with a Failed Zero Forcing Number of Two

1
Department of Mathematical Sciences, University of Arkansas, Fayetteville, AR 72701, USA
2
Mathematics Department, California State University—Dominguez Hills, Carson, CA 90747, USA
3
Department of Mathematics, Southern Methodist University, Dallas, TX 75205, USA
4
School of Mathematical Sciences, Rochester Institute of Technology, Rochester, NY 14623, USA
*
Author to whom correspondence should be addressed.
Academic Editors: Ioan Rașa and Haruo Hosoya
Symmetry 2021, 13(11), 2221; https://doi.org/10.3390/sym13112221
Received: 18 October 2021 / Revised: 9 November 2021 / Accepted: 17 November 2021 / Published: 20 November 2021
(This article belongs to the Special Issue Graph Algorithms and Graph Theory)
Given a graph G, the zero forcing number of G, Z(G), is the smallest cardinality of any set S of vertices on which repeated applications of the forcing rule results in all vertices being in S. The forcing rule is: if a vertex v is in S, and exactly one neighbor u of v is not in S, then u is added to S in the next iteration. Zero forcing numbers have attracted great interest over the past 15 years and have been well studied. In this paper, we investigate the largest size of a set S that does not force all of the vertices in a graph to be in S. This quantity is known as the failed zero forcing number of a graph and will be denoted by F(G). We present new results involving this parameter. In particular, we completely characterize all graphs G where F(G)=2, solving a problem posed in 2015 by Fetcie, Jacob, and Saavedra. View Full-Text
Keywords: failed zero forcing number; zero forcing number; graph labelling failed zero forcing number; zero forcing number; graph labelling
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MDPI and ACS Style

Gomez, L.; Rubi, K.; Terrazas, J.; Narayan, D.A. All Graphs with a Failed Zero Forcing Number of Two. Symmetry 2021, 13, 2221. https://doi.org/10.3390/sym13112221

AMA Style

Gomez L, Rubi K, Terrazas J, Narayan DA. All Graphs with a Failed Zero Forcing Number of Two. Symmetry. 2021; 13(11):2221. https://doi.org/10.3390/sym13112221

Chicago/Turabian Style

Gomez, Luis, Karla Rubi, Jorden Terrazas, and Darren A. Narayan. 2021. "All Graphs with a Failed Zero Forcing Number of Two" Symmetry 13, no. 11: 2221. https://doi.org/10.3390/sym13112221

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