{"title":"具有树状结构分解的图的孪生宽度","authors":"Irene Heinrich, Simon Rassmann","doi":"10.4230/LIPIcs.IPEC.2023.25","DOIUrl":null,"url":null,"abstract":"The twin-width of a graph measures its distance to co-graphs and generalizes classical width concepts such as tree-width or rank-width. Since its introduction in 2020 (Bonnet et. al. 2020), a mass of new results has appeared relating twin width to group theory, model theory, combinatorial optimization, and structural graph theory. We take a detailed look at the interplay between the twin-width of a graph and the twin-width of its components under tree-structured decompositions: We prove that the twin-width of a graph is at most twice its strong tree-width, contrasting nicely with the result of (Bonnet and D\\'epr\\'es 2022), which states that twin-width can be exponential in tree-width. Further, we employ the fundamental concept from structural graph theory of decomposing a graph into highly connected components, in order to obtain an optimal linear bound on the twin-width of a graph given the widths of its biconnected components. For triconnected components we obtain a linear upper bound if we add red edges to the components indicating the splits which led to the components. Extending this approach to quasi-4-connectivity, we obtain a quadratic upper bound. Finally, we investigate how the adhesion of a tree decomposition influences the twin-width of the decomposed graph.","PeriodicalId":137775,"journal":{"name":"International Symposium on Parameterized and Exact Computation","volume":"32 1","pages":"25:1-25:17"},"PeriodicalIF":0.0000,"publicationDate":"2023-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Twin-Width of Graphs with Tree-Structured Decompositions\",\"authors\":\"Irene Heinrich, Simon Rassmann\",\"doi\":\"10.4230/LIPIcs.IPEC.2023.25\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The twin-width of a graph measures its distance to co-graphs and generalizes classical width concepts such as tree-width or rank-width. Since its introduction in 2020 (Bonnet et. al. 2020), a mass of new results has appeared relating twin width to group theory, model theory, combinatorial optimization, and structural graph theory. We take a detailed look at the interplay between the twin-width of a graph and the twin-width of its components under tree-structured decompositions: We prove that the twin-width of a graph is at most twice its strong tree-width, contrasting nicely with the result of (Bonnet and D\\\\'epr\\\\'es 2022), which states that twin-width can be exponential in tree-width. Further, we employ the fundamental concept from structural graph theory of decomposing a graph into highly connected components, in order to obtain an optimal linear bound on the twin-width of a graph given the widths of its biconnected components. For triconnected components we obtain a linear upper bound if we add red edges to the components indicating the splits which led to the components. Extending this approach to quasi-4-connectivity, we obtain a quadratic upper bound. Finally, we investigate how the adhesion of a tree decomposition influences the twin-width of the decomposed graph.\",\"PeriodicalId\":137775,\"journal\":{\"name\":\"International Symposium on Parameterized and Exact Computation\",\"volume\":\"32 1\",\"pages\":\"25:1-25:17\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-08-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"International Symposium on Parameterized and Exact Computation\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.4230/LIPIcs.IPEC.2023.25\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Symposium on Parameterized and Exact Computation","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4230/LIPIcs.IPEC.2023.25","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
图的孪生宽度(twin-width)测量图与共图间的距离,是对树宽或秩宽等经典宽度概念的概括。自 2020 年提出(Bonnet 等人,2020 年)以来,出现了大量与群论、模型论、组合优化和结构图理论相关的新成果。我们将详细研究在树状结构分解下,图的孪生宽度与其成分的孪生宽度之间的相互作用:我们证明了图的孪生宽最多是其强树宽的两倍,这与(Bonnet and D\'epr\'es 2022)的结果形成了鲜明对比,后者指出孪生宽可以是树宽的指数。此外,我们还运用了结构图理论中的基本概念--将图分解为高连接成分,从而在给定双连接成分宽度的情况下,获得图的孪生宽度的最优线性约束。对于三连接成分,如果我们在成分上添加红色边,表示导致成分分裂的分裂点,就能得到线性上界。将这种方法扩展到准 4 连通性,我们会得到一个二次方上界。最后,我们研究了树形分解的附着力如何影响分解图的孪生宽度。
Twin-Width of Graphs with Tree-Structured Decompositions
The twin-width of a graph measures its distance to co-graphs and generalizes classical width concepts such as tree-width or rank-width. Since its introduction in 2020 (Bonnet et. al. 2020), a mass of new results has appeared relating twin width to group theory, model theory, combinatorial optimization, and structural graph theory. We take a detailed look at the interplay between the twin-width of a graph and the twin-width of its components under tree-structured decompositions: We prove that the twin-width of a graph is at most twice its strong tree-width, contrasting nicely with the result of (Bonnet and D\'epr\'es 2022), which states that twin-width can be exponential in tree-width. Further, we employ the fundamental concept from structural graph theory of decomposing a graph into highly connected components, in order to obtain an optimal linear bound on the twin-width of a graph given the widths of its biconnected components. For triconnected components we obtain a linear upper bound if we add red edges to the components indicating the splits which led to the components. Extending this approach to quasi-4-connectivity, we obtain a quadratic upper bound. Finally, we investigate how the adhesion of a tree decomposition influences the twin-width of the decomposed graph.