Pablo Blanco, Linda Cook, Meike Hatzel, Claire Hilaire, Freddie Illingworth, Rose McCarty
{"title":"关于树为未成年人的树分解","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":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"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\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"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\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23083\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23083","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","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 .
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