{"title":"混合图中2-顶点连通方向的复杂性","authors":"Florian Hörsch , Zoltán Szigeti","doi":"10.1016/j.disopt.2023.100774","DOIUrl":null,"url":null,"abstract":"<div><p>We consider two possible extensions of a theorem of Thomassen characterizing the graphs admitting a 2-vertex-connected orientation. First, we show that the problem of deciding whether a mixed graph has a 2-vertex-connected orientation is NP-hard. This answers a question of Bang-Jensen, Huang and Zhu. For the second part, we call a directed graph <span><math><mrow><mi>D</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>A</mi><mo>)</mo></mrow></mrow></math></span>\n<span><math><mrow><mn>2</mn><mi>T</mi></mrow></math></span>-connected for some <span><math><mrow><mi>T</mi><mo>⊆</mo><mi>V</mi></mrow></math></span> if <span><math><mi>D</mi></math></span> is 2-arc-connected and <span><math><mrow><mi>D</mi><mo>−</mo><mi>v</mi></mrow></math></span> is strongly connected for all <span><math><mrow><mi>v</mi><mo>∈</mo><mi>T</mi></mrow></math></span>. We deduce a characterization of the graphs admitting a <span><math><mrow><mn>2</mn><mi>T</mi></mrow></math></span>-connected orientation from the theorem of Thomassen.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The complexity of 2-vertex-connected orientation in mixed graphs\",\"authors\":\"Florian Hörsch , Zoltán Szigeti\",\"doi\":\"10.1016/j.disopt.2023.100774\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We consider two possible extensions of a theorem of Thomassen characterizing the graphs admitting a 2-vertex-connected orientation. First, we show that the problem of deciding whether a mixed graph has a 2-vertex-connected orientation is NP-hard. This answers a question of Bang-Jensen, Huang and Zhu. For the second part, we call a directed graph <span><math><mrow><mi>D</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>A</mi><mo>)</mo></mrow></mrow></math></span>\\n<span><math><mrow><mn>2</mn><mi>T</mi></mrow></math></span>-connected for some <span><math><mrow><mi>T</mi><mo>⊆</mo><mi>V</mi></mrow></math></span> if <span><math><mi>D</mi></math></span> is 2-arc-connected and <span><math><mrow><mi>D</mi><mo>−</mo><mi>v</mi></mrow></math></span> is strongly connected for all <span><math><mrow><mi>v</mi><mo>∈</mo><mi>T</mi></mrow></math></span>. We deduce a characterization of the graphs admitting a <span><math><mrow><mn>2</mn><mi>T</mi></mrow></math></span>-connected orientation from the theorem of Thomassen.</p></div>\",\"PeriodicalId\":50571,\"journal\":{\"name\":\"Discrete Optimization\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2023-05-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Discrete Optimization\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S1572528623000166\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Optimization","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1572528623000166","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
The complexity of 2-vertex-connected orientation in mixed graphs
We consider two possible extensions of a theorem of Thomassen characterizing the graphs admitting a 2-vertex-connected orientation. First, we show that the problem of deciding whether a mixed graph has a 2-vertex-connected orientation is NP-hard. This answers a question of Bang-Jensen, Huang and Zhu. For the second part, we call a directed graph
-connected for some if is 2-arc-connected and is strongly connected for all . We deduce a characterization of the graphs admitting a -connected orientation from the theorem of Thomassen.
期刊介绍:
Discrete Optimization publishes research papers on the mathematical, computational and applied aspects of all areas of integer programming and combinatorial optimization. In addition to reports on mathematical results pertinent to discrete optimization, the journal welcomes submissions on algorithmic developments, computational experiments, and novel applications (in particular, large-scale and real-time applications). The journal also publishes clearly labelled surveys, reviews, short notes, and open problems. Manuscripts submitted for possible publication to Discrete Optimization should report on original research, should not have been previously published, and should not be under consideration for publication by any other journal.