{"title":"平面图形中产品结构的优化算法","authors":"P. Bose, Pat Morin, Saeed Odak","doi":"10.4230/LIPIcs.SWAT.2022.19","DOIUrl":null,"url":null,"abstract":"The \\emph{Product Structure Theorem} for planar graphs (Dujmovi\\'c et al.\\ \\emph{JACM}, \\textbf{67}(4):22) states that any planar graph is contained in the strong product of a planar $3$-tree, a path, and a $3$-cycle. We give a simple linear-time algorithm for finding this decomposition as well as several related decompositions. This improves on the previous $O(n\\log n)$ time algorithm (Morin.\\ \\emph{Algorithmica}, \\textbf{85}(5):1544--1558).","PeriodicalId":447445,"journal":{"name":"Scandinavian Workshop on Algorithm Theory","volume":"53 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2022-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"9","resultStr":"{\"title\":\"An Optimal Algorithm for Product Structure in Planar Graphs\",\"authors\":\"P. Bose, Pat Morin, Saeed Odak\",\"doi\":\"10.4230/LIPIcs.SWAT.2022.19\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The \\\\emph{Product Structure Theorem} for planar graphs (Dujmovi\\\\'c et al.\\\\ \\\\emph{JACM}, \\\\textbf{67}(4):22) states that any planar graph is contained in the strong product of a planar $3$-tree, a path, and a $3$-cycle. We give a simple linear-time algorithm for finding this decomposition as well as several related decompositions. This improves on the previous $O(n\\\\log n)$ time algorithm (Morin.\\\\ \\\\emph{Algorithmica}, \\\\textbf{85}(5):1544--1558).\",\"PeriodicalId\":447445,\"journal\":{\"name\":\"Scandinavian Workshop on Algorithm Theory\",\"volume\":\"53 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2022-02-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"9\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Scandinavian Workshop on Algorithm Theory\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.4230/LIPIcs.SWAT.2022.19\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Scandinavian Workshop on Algorithm Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4230/LIPIcs.SWAT.2022.19","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 9
摘要
平面图的\emph{积结构定理}(dujmovovic et al.)\emph{JACM}, \textbf{67}(4):22)指出任何平面图都包含在平面$3$ -tree、路径和$3$ -cycle的强积中。我们给出了一个简单的线性时间算法来找到这个分解，以及几个相关的分解。这改进了以前的$O(n\log n)$时间算法(Morin。\emph{算法}，\textbf{85}(5):1544—1558。
An Optimal Algorithm for Product Structure in Planar Graphs
The \emph{Product Structure Theorem} for planar graphs (Dujmovi\'c et al.\ \emph{JACM}, \textbf{67}(4):22) states that any planar graph is contained in the strong product of a planar $3$-tree, a path, and a $3$-cycle. We give a simple linear-time algorithm for finding this decomposition as well as several related decompositions. This improves on the previous $O(n\log n)$ time algorithm (Morin.\ \emph{Algorithmica}, \textbf{85}(5):1544--1558).
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