{"title":"The minimum number of maximal independent sets in graphs with given order and independence number","authors":"Yuting Tian, Jianhua Tu","doi":"10.1016/j.dam.2025.02.027","DOIUrl":null,"url":null,"abstract":"<div><div>Let <span><math><mrow><mi>M</mi><mi>I</mi><mi>S</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> be the set of all maximal independent sets in a graph <span><math><mi>G</mi></math></span>, and let <span><math><mrow><mi>m</mi><mi>i</mi><mi>s</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>|</mo><mi>M</mi><mi>I</mi><mi>S</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>|</mo></mrow></mrow></math></span>. In this paper, we show that for any tree <span><math><mi>T</mi></math></span> with <span><math><mi>n</mi></math></span> vertices and independence number <span><math><mi>α</mi></math></span>, <span><span><span><math><mrow><mi>m</mi><mi>i</mi><mi>s</mi><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mo>≥</mo><mi>f</mi><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mi>α</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow><mo>,</mo></mrow></math></span></span></span>and for any unicyclic graph <span><math><mi>G</mi></math></span> with <span><math><mi>n</mi></math></span> vertices and independence number <span><math><mi>α</mi></math></span>, <span><span><span><math><mrow><mi>m</mi><mi>i</mi><mi>s</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><mfenced><mrow><mtable><mtr><mtd><mn>2</mn><mo>,</mo><mspace></mspace></mtd><mtd><mtext>if</mtext><mspace></mspace><mi>n</mi><mo>=</mo><mn>4</mn><mo>,</mo></mtd></mtr><mtr><mtd><mn>3</mn><mo>,</mo><mspace></mspace></mtd><mtd><mtext>if</mtext><mspace></mspace><mi>α</mi><mo>=</mo><mi>n</mi><mo>−</mo><mn>2</mn><mspace></mspace><mtext>and</mtext><mspace></mspace><mi>n</mi><mo>≠</mo><mn>4</mn><mo>,</mo></mtd></mtr><mtr><mtd><mn>2</mn><mi>f</mi><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mi>α</mi><mo>)</mo></mrow><mo>,</mo><mspace></mspace></mtd><mtd><mtext>if</mtext><mspace></mspace><mi>n</mi><mo>≥</mo><mn>6</mn><mspace></mspace><mtext>and</mtext><mspace></mspace><mrow><mo>⌈</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></mrow><mo>≤</mo><mi>α</mi><mo><</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo>,</mo></mtd></mtr><mtr><mtd><mi>f</mi><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mi>α</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow><mo>−</mo><mi>f</mi><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mi>α</mi><mo>−</mo><mn>3</mn><mo>)</mo></mrow><mo>,</mo><mspace></mspace></mtd><mtd><mtext>if</mtext><mspace></mspace><mi>n</mi><mo>≥</mo><mn>5</mn><mo>,</mo><mspace></mspace><mtext>and</mtext><mspace></mspace><mi>n</mi><mspace></mspace><mtext>is odd</mtext><mo>,</mo><mspace></mspace><mtext>and</mtext><mspace></mspace><mi>α</mi><mo>=</mo><mrow><mo>⌊</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌋</mo></mrow><mo>,</mo></mtd></mtr></mtable></mrow></mfenced></mrow></math></span></span></span>where <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> represents the <span><math><mi>n</mi></math></span>th Fibonacci number. Moreover, we also show that the above inequalities are sharp.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"368 ","pages":"Pages 52-65"},"PeriodicalIF":1.0000,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0166218X25000988","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
Let be the set of all maximal independent sets in a graph , and let . In this paper, we show that for any tree with vertices and independence number , and for any unicyclic graph with vertices and independence number , where represents the th Fibonacci number. Moreover, we also show that the above inequalities are sharp.
期刊介绍:
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