{"title":"燃烧杠铃的数量","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":6951,"journal":{"name":"Acta Mathematicae Applicatae Sinica, English Series","volume":"40 2","pages":"526 - 538"},"PeriodicalIF":0.9000,"publicationDate":"2024-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"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\":6951,\"journal\":{\"name\":\"Acta Mathematicae Applicatae Sinica, English Series\",\"volume\":\"40 2\",\"pages\":\"526 - 538\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-03-27\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Acta Mathematicae Applicatae Sinica, English Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10255-024-1113-8\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematicae Applicatae Sinica, English Series","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10255-024-1113-8","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
摘要
图 G 的燃烧数 b(G)定义为：对于图 G 的每个顶点 u，存在 i∈ {1,...,k}，且 dG(u,xi)≤k-i，以及 dG(xi,xj)≥j-i（对于任意 1≤i <j≤k）的最小整数 k，对于该整数，存在顶点 x1,...,xk，且对于图 G 的每个顶点 u，存在 i∈ {1,...,k}，且 dG(u,xi)≤k-i。即使对于某些最大阶数为三的无循环图，图燃烧问题也被证明是 NP-完全的。在本文中，我们将分别确定所有短杠铃和长杠铃的燃烧数。
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.
期刊介绍：
Acta Mathematicae Applicatae Sinica (English Series) is a quarterly journal established by the Chinese Mathematical Society. The journal publishes high quality research papers from all branches of applied mathematics, and particularly welcomes those from partial differential equations, computational mathematics, applied probability, mathematical finance, statistics, dynamical systems, optimization and management science.
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