{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:33:39Z","timestamp":1760236419701,"version":"build-2065373602"},"reference-count":8,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2021,11,20]],"date-time":"2021-11-20T00:00:00Z","timestamp":1637366400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100008982","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1950189"],"award-info":[{"award-number":["1950189"]}],"id":[{"id":"10.13039\/501100008982","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.3390\/sym13112221","type":"journal-article","created":{"date-parts":[[2021,11,21]],"date-time":"2021-11-21T21:00:50Z","timestamp":1637528450000},"page":"2221","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["All Graphs with a Failed Zero Forcing Number of Two"],"prefix":"10.3390","volume":"13","author":[{"given":"Luis","family":"Gomez","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, University of Arkansas, Fayetteville, AR 72701, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Karla","family":"Rubi","sequence":"additional","affiliation":[{"name":"Mathematics Department, California State University\u2014Dominguez Hills, Carson, CA 90747, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jorden","family":"Terrazas","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Southern Methodist University, Dallas, TX 75205, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Darren A.","family":"Narayan","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Rochester Institute of Technology, Rochester, NY 14623, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,11,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"558","DOI":"10.1016\/j.laa.2007.05.036","article-title":"The minimum rank of symmetric matrices described by a graph: A survey","volume":"426","author":"Fallat","year":"2007","journal-title":"Linear Algebra Appl."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"99","DOI":"10.2140\/involve.2015.8.99","article-title":"The Failed Zero-Forcing Number of a Graph","volume":"8","author":"Fetcie","year":"2015","journal-title":"Involve"},{"key":"ref_3","unstructured":"Adams, A., and Jacob, B. (2019). Failed zero forcing and critical sets on directed graphs. arXiv."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"102","DOI":"10.1016\/j.tcs.2016.11.032","article-title":"On the complexity of failed zero forcing","volume":"660","author":"Shitov","year":"2017","journal-title":"Theor. Comput. Sci."},{"key":"ref_5","first-page":"380","article-title":"On the zero blocking number of rectangular, cylindrical, and M\u00f6bius grids","volume":"285","author":"Crawford","year":"2020","journal-title":"Discret. Appl. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"380","DOI":"10.1016\/j.dam.2020.06.002","article-title":"Blocking zero forcing processes in Cartesian products of graphs","volume":"285","author":"Karst","year":"2020","journal-title":"Discret. Appl. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1007\/BF01895716","article-title":"Asymmetric Graphs","volume":"14","author":"Erdos","year":"1963","journal-title":"Acta Math. Acad. Sci. Hungar."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/0012-365X(84)90033-5","article-title":"A table of connected graphs on six vertices","volume":"50","year":"1984","journal-title":"Discret. Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/11\/2221\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:33:17Z","timestamp":1760167997000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/11\/2221"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,11,20]]},"references-count":8,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2021,11]]}},"alternative-id":["sym13112221"],"URL":"https:\/\/doi.org\/10.3390\/sym13112221","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,11,20]]}}}