{"title":"Burning Numbers of Barbells","authors":"Hui-qing Liu, Rui-ting Zhang, Xiao-lan Hu","doi":"10.1007/s10255-024-1113-8","DOIUrl":null,"url":null,"abstract":"<div><p>Motivated by a discrete-time process intended to measure the speed of the spread of contagion in a graph, the burning number <i>b</i>(<i>G</i>) of a graph <i>G</i>, is defined as the smallest integer <i>k</i> for which there are vertices <i>x</i><sub>1</sub>,…,<i>x</i><sub><i>k</i></sub> such that for every vertex <i>u</i> of <i>G</i>, there exists <i>i</i> ∈ {1,…,<i>k</i>} with <i>d</i><sub><i>G</i></sub>(<i>u, x</i><sub><i>i</i></sub>) ≤ <i>k</i> − <i>i</i>, and <i>d</i><sub><i>G</i></sub>(<i>x</i><sub><i>i</i></sub>, <i>x</i><sub><i>j</i></sub>) ≥ <i>j</i> − <i>i</i> for any 1 ≤ <i>i</i> < <i>j</i> ≤ <i>k</i>. The graph burning problem has been shown to be NP-complete even for some acyclic graphs with maximum degree three. In this paper, we determine the burning numbers of all short barbells and long barbells, respectively.</p></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10255-024-1113-8","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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Abstract
Motivated by a discrete-time process intended to measure the speed of the spread of contagion in a graph, the burning number b(G) of a graph G, is defined as the smallest integer k for which there are vertices x1,…,xk such that for every vertex u of G, there exists i ∈ {1,…,k} with dG(u, xi) ≤ k − i, and dG(xi, xj) ≥ j − i for any 1 ≤ i < j ≤ k. The graph burning problem has been shown to be NP-complete even for some acyclic graphs with maximum degree three. In this paper, we determine the burning numbers of all short barbells and long barbells, respectively.
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