{"title":"Improved bounds on the cop number when forbidding a minor","authors":"Franklin Kenter, Erin Meger, Jérémie Turcotte","doi":"10.1002/jgt.23194","DOIUrl":null,"url":null,"abstract":"<p>Andreae proved that the cop number of connected <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>H</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23194:jgt23194-math-0001\" wiley:location=\"equation/jgt23194-math-0001.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\n </semantics></math>-minor-free graphs is bounded for every graph <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>H</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23194:jgt23194-math-0002\" wiley:location=\"equation/jgt23194-math-0002.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\n </semantics></math>. In particular, the cop number is at most <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mo>∣</mo>\n \n <mi>E</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>H</mi>\n \n <mo>−</mo>\n \n <mi>h</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n \n <mo>∣</mo>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23194:jgt23194-math-0003\" wiley:location=\"equation/jgt23194-math-0003.png\"><mrow><mrow><mo>\\unicode{x02223}</mo><mi>E</mi><mrow><mo>(</mo><mrow><mi>H</mi><mo>\\unicode{x02212}</mo><mi>h</mi></mrow><mo>)</mo></mrow><mo>\\unicode{x02223}</mo></mrow></mrow></math></annotation>\n </semantics></math> if <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>H</mi>\n \n <mo>−</mo>\n \n <mi>h</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23194:jgt23194-math-0004\" wiley:location=\"equation/jgt23194-math-0004.png\"><mrow><mrow><mi>H</mi><mo>\\unicode{x02212}</mo><mi>h</mi></mrow></mrow></math></annotation>\n </semantics></math> contains no isolated vertex, where <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>h</mi>\n \n <mo>∈</mo>\n \n <mi>V</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>H</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23194:jgt23194-math-0005\" wiley:location=\"equation/jgt23194-math-0005.png\"><mrow><mrow><mi>h</mi><mo>\\unicode{x02208}</mo><mi>V</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math>. The main result of this paper is an improvement on this bound, which is most significant when <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>H</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23194:jgt23194-math-0006\" wiley:location=\"equation/jgt23194-math-0006.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\n </semantics></math> is small or sparse, for instance, when <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>H</mi>\n \n <mo>−</mo>\n \n <mi>h</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23194:jgt23194-math-0007\" wiley:location=\"equation/jgt23194-math-0007.png\"><mrow><mrow><mi>H</mi><mo>\\unicode{x02212}</mo><mi>h</mi></mrow></mrow></math></annotation>\n </semantics></math> can be obtained from another graph by multiple edge subdivisions. Some consequences of this result are improvements on the upper bound for the cop number of <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>K</mi>\n \n <mrow>\n <mn>3</mn>\n \n <mo>,</mo>\n \n <mi>t</mi>\n </mrow>\n </msub>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23194:jgt23194-math-0008\" wiley:location=\"equation/jgt23194-math-0008.png\"><mrow><mrow><msub><mi>K</mi><mrow><mn>3</mn><mo>,</mo><mi>t</mi></mrow></msub></mrow></mrow></math></annotation>\n </semantics></math>-minor-free graphs, <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>K</mi>\n \n <mrow>\n <mn>2</mn>\n \n <mo>,</mo>\n \n <mi>t</mi>\n </mrow>\n </msub>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23194:jgt23194-math-0009\" wiley:location=\"equation/jgt23194-math-0009.png\"><mrow><mrow><msub><mi>K</mi><mrow><mn>2</mn><mo>,</mo><mi>t</mi></mrow></msub></mrow></mrow></math></annotation>\n </semantics></math>-minor-free graphs and linklessly embeddable graphs.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 3","pages":"620-646"},"PeriodicalIF":0.9000,"publicationDate":"2024-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23194","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
Andreae proved that the cop number of connected -minor-free graphs is bounded for every graph . In particular, the cop number is at most if contains no isolated vertex, where . The main result of this paper is an improvement on this bound, which is most significant when is small or sparse, for instance, when can be obtained from another graph by multiple edge subdivisions. Some consequences of this result are improvements on the upper bound for the cop number of -minor-free graphs, -minor-free graphs and linklessly embeddable graphs.
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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.
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