{"title":"在树宽度2的图形上的电话广播","authors":"Prafullkumar Tale","doi":"10.1016/j.tcs.2025.115282","DOIUrl":null,"url":null,"abstract":"<div><div>Consider the <span>Telephone Broadcast</span> problem in which an input is a connected graph <em>G</em> on <em>n</em> vertices, a source vertex <span><math><mi>s</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, and a positive integer <em>t</em>. The objective is to decide whether there is a broadcast protocol from <em>s</em> that ensures that all the vertices of <em>G</em> get the message in at most <em>t</em> rounds. We consider the broadcast protocol where, in a round, any node aware of the message can forward it to at most one of its neighbors. Fomin, Fraigniaud, and Golovach [WG 2023; TCS 2024] asked whether the problem is <figure><img></figure> when parameterized by the feedback vertex set number of the graph. We answer this question in the negative.<ul><li><span>•</span><span><div><span>Telephone Broadcast</span>, when restricted to graphs of the feedback vertex number one, and hence treewidth of two, is <figure><img></figure>-<span>complete</span>.</div></span></li></ul> We find this a relatively rare example of problems that admit a polynomial-time algorithm on trees but is <figure><img></figure>-<span>complete</span> on graphs of treewidth two.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1045 ","pages":"Article 115282"},"PeriodicalIF":1.0000,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Telephone Broadcast on graphs of treewidth two\",\"authors\":\"Prafullkumar Tale\",\"doi\":\"10.1016/j.tcs.2025.115282\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>Consider the <span>Telephone Broadcast</span> problem in which an input is a connected graph <em>G</em> on <em>n</em> vertices, a source vertex <span><math><mi>s</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, and a positive integer <em>t</em>. The objective is to decide whether there is a broadcast protocol from <em>s</em> that ensures that all the vertices of <em>G</em> get the message in at most <em>t</em> rounds. We consider the broadcast protocol where, in a round, any node aware of the message can forward it to at most one of its neighbors. Fomin, Fraigniaud, and Golovach [WG 2023; TCS 2024] asked whether the problem is <figure><img></figure> when parameterized by the feedback vertex set number of the graph. We answer this question in the negative.<ul><li><span>•</span><span><div><span>Telephone Broadcast</span>, when restricted to graphs of the feedback vertex number one, and hence treewidth of two, is <figure><img></figure>-<span>complete</span>.</div></span></li></ul> We find this a relatively rare example of problems that admit a polynomial-time algorithm on trees but is <figure><img></figure>-<span>complete</span> on graphs of treewidth two.</div></div>\",\"PeriodicalId\":49438,\"journal\":{\"name\":\"Theoretical Computer Science\",\"volume\":\"1045 \",\"pages\":\"Article 115282\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2025-05-06\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Theoretical Computer Science\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0304397525002208\",\"RegionNum\":4,\"RegionCategory\":\"计算机科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Theoretical Computer Science","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0304397525002208","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
摘要
考虑电话广播问题，其中输入是n个顶点的连通图G，源顶点s∈V(G)和正整数t。目标是确定是否存在一个来自s的广播协议，以确保G的所有顶点最多在t轮中获得消息。我们考虑广播协议，在一轮中，任何知道消息的节点最多可以将其转发给它的一个邻居。Fomin, Fraigniaud, and Golovach [WG 2023；TCS 2024]询问问题是否在图的反馈顶点集数参数化时出现。我们的回答是否定的。•电话广播，当被限制到反馈顶点数为1的图时，因此树宽为2，是-完整的。我们发现这是一个相对罕见的例子，在树上允许多项式时间算法，但在树宽为2的图上是不完全的。
Consider the Telephone Broadcast problem in which an input is a connected graph G on n vertices, a source vertex , and a positive integer t. The objective is to decide whether there is a broadcast protocol from s that ensures that all the vertices of G get the message in at most t rounds. We consider the broadcast protocol where, in a round, any node aware of the message can forward it to at most one of its neighbors. Fomin, Fraigniaud, and Golovach [WG 2023; TCS 2024] asked whether the problem is when parameterized by the feedback vertex set number of the graph. We answer this question in the negative.
•
Telephone Broadcast, when restricted to graphs of the feedback vertex number one, and hence treewidth of two, is -complete.
We find this a relatively rare example of problems that admit a polynomial-time algorithm on trees but is -complete on graphs of treewidth two.
期刊介绍：
Theoretical Computer Science is mathematical and abstract in spirit, but it derives its motivation from practical and everyday computation. Its aim is to understand the nature of computation and, as a consequence of this understanding, provide more efficient methodologies. All papers introducing or studying mathematical, logic and formal concepts and methods are welcome, provided that their motivation is clearly drawn from the field of computing.
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