Pablo Blanco, Linda Cook, Meike Hatzel, Claire Hilaire, Freddie Illingworth, Rose McCarty
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We prove that this is false, even when <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> has treewidth 2 and <span></span><math>\n <semantics>\n <mrow>\n <mi>T</mi>\n </mrow>\n <annotation> $T$</annotation>\n </semantics></math> is allowed to be a minor of <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math>.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"106 2","pages":"296-306"},"PeriodicalIF":1.0000,"publicationDate":"2024-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23083","citationCount":"0","resultStr":"{\"title\":\"On tree decompositions whose trees are minors\",\"authors\":\"Pablo Blanco, Linda Cook, Meike Hatzel, Claire Hilaire, Freddie Illingworth, Rose McCarty\",\"doi\":\"10.1002/jgt.23083\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In 2019, Dvořák asked whether every connected graph <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n <annotation> $G$</annotation>\\n </semantics></math> has a tree decomposition <span></span><math>\\n <semantics>\\n <mrow>\\n <mo>(</mo>\\n \\n <mrow>\\n <mi>T</mi>\\n \\n <mo>,</mo>\\n \\n <mi>B</mi>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n <annotation> $(T,{\\\\rm{ {\\\\mathcal B} }})$</annotation>\\n </semantics></math> so that <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>T</mi>\\n </mrow>\\n <annotation> $T$</annotation>\\n </semantics></math> is a subgraph of <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n <annotation> $G$</annotation>\\n </semantics></math> and the width of <span></span><math>\\n <semantics>\\n <mrow>\\n <mo>(</mo>\\n \\n <mrow>\\n <mi>T</mi>\\n \\n <mo>,</mo>\\n \\n <mi>B</mi>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n <annotation> $(T,{\\\\rm{ {\\\\mathcal B} }})$</annotation>\\n </semantics></math> is bounded by a function of the treewidth of <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n <annotation> $G$</annotation>\\n </semantics></math>. We prove that this is false, even when <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n <annotation> $G$</annotation>\\n </semantics></math> has treewidth 2 and <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>T</mi>\\n </mrow>\\n <annotation> $T$</annotation>\\n </semantics></math> is allowed to be a minor of <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n <annotation> $G$</annotation>\\n </semantics></math>.</p>\",\"PeriodicalId\":16014,\"journal\":{\"name\":\"Journal of Graph Theory\",\"volume\":\"106 2\",\"pages\":\"296-306\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-02-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23083\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Graph Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23083\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23083","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
In 2019, Dvořák asked whether every connected graph has a tree decomposition so that is a subgraph of and the width of is bounded by a function of the treewidth of . We prove that this is false, even when has treewidth 2 and is allowed to be a minor of .
期刊介绍：
The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences.
A subscription to the Journal of Graph Theory includes a subscription to the Journal of Combinatorial Designs .
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