{"title":"Hendrickson关于全局刚性图猜想的极小反例","authors":"Georg Grasegger","doi":"10.1016/j.exco.2023.100106","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper we consider the class of graphs which are redundantly <span><math><mi>d</mi></math></span>-rigid and <span><math><mrow><mo>(</mo><mi>d</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></math></span>-connected but not globally <span><math><mi>d</mi></math></span>-rigid, where <span><math><mi>d</mi></math></span> is the dimension. This class arises from counterexamples to a conjecture by Bruce Hendrickson. It seems that there are relatively few graphs in this class for a given number of vertices. Using computations we show that <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>5</mn><mo>,</mo><mn>5</mn></mrow></msub></math></span> is indeed the smallest counterexample to the conjecture.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"3 ","pages":"Article 100106"},"PeriodicalIF":0.6000,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Minimal counterexamples to Hendrickson’s conjecture on globally rigid graphs\",\"authors\":\"Georg Grasegger\",\"doi\":\"10.1016/j.exco.2023.100106\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this paper we consider the class of graphs which are redundantly <span><math><mi>d</mi></math></span>-rigid and <span><math><mrow><mo>(</mo><mi>d</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></math></span>-connected but not globally <span><math><mi>d</mi></math></span>-rigid, where <span><math><mi>d</mi></math></span> is the dimension. This class arises from counterexamples to a conjecture by Bruce Hendrickson. It seems that there are relatively few graphs in this class for a given number of vertices. Using computations we show that <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>5</mn><mo>,</mo><mn>5</mn></mrow></msub></math></span> is indeed the smallest counterexample to the conjecture.</p></div>\",\"PeriodicalId\":100517,\"journal\":{\"name\":\"Examples and Counterexamples\",\"volume\":\"3 \",\"pages\":\"Article 100106\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2023-11-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Examples and Counterexamples\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S2666657X23000083\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"2023/4/6 0:00:00\",\"PubModel\":\"Epub\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Examples and Counterexamples","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S2666657X23000083","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2023/4/6 0:00:00","PubModel":"Epub","JCR":"","JCRName":"","Score":null,"Total":0}
Minimal counterexamples to Hendrickson’s conjecture on globally rigid graphs
In this paper we consider the class of graphs which are redundantly -rigid and -connected but not globally -rigid, where is the dimension. This class arises from counterexamples to a conjecture by Bruce Hendrickson. It seems that there are relatively few graphs in this class for a given number of vertices. Using computations we show that is indeed the smallest counterexample to the conjecture.
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