{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T02:15:19Z","timestamp":1772504119910,"version":"3.50.1"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Bootstrap percolation, one of the simplest cellular automata, can be seen as a model of the spread of infection. In $r$-neighbour bootstrap percolation on a graph $G$ we assign a state, infected or healthy, to every vertex of $G$ and then update these states in successive rounds, according to the following simple local update rule: infected vertices of $G$ remain infected forever and a healthy vertex becomes infected if it has at least $r$ already infected neighbours. We say that percolation occurs if eventually every vertex of $G$ becomes infected.\u00a0A well known and celebrated fact about the classical model of $2$-neighbour bootstrap percolation on the $n \\times n$ square grid is that the smallest size of an initially infected set which percolates in this process is $n$. In this paper we consider the problem of finding the maximum time a $2$-neighbour bootstrap process on $[n]^2$ with $n$ initially infected vertices can take to eventually infect the entire vertex set. Answering a question posed by Bollob\u00e1s we compute the exact value for this maximum showing that, for $n \\ge 4$, it is equal to the integer nearest to $(5n^2-2n)\/8$.<\/jats:p>","DOI":"10.37236\/2542","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T01:32:09Z","timestamp":1578706329000},"source":"Crossref","is-referenced-by-count":17,"title":["On Slowly Percolating Sets of Minimal Size in Bootstrap Percolation"],"prefix":"10.37236","volume":"20","author":[{"given":"Fabricio","family":"Benevides","sequence":"first","affiliation":[]},{"given":"Micha\u0142","family":"Przykucki","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2013,6,7]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i2p46\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i2p46\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T11:19:06Z","timestamp":1579259946000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v20i2p46"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,6,7]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2013,4,9]]}},"URL":"https:\/\/doi.org\/10.37236\/2542","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,6,7]]},"article-number":"P46"}}