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All Graphs with a Failed Zero Forcing Number of Two

Department of Mathematical Sciences, University of Arkansas, Fayetteville, AR 72701, USA
Mathematics Department, California State University—Dominguez Hills, Carson, CA 90747, USA
Department of Mathematics, Southern Methodist University, Dallas, TX 75205, USA
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;
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.

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.

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.

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