Noga Alon, Emil Powierski, Michael Savery, Alex Scott, Elizabeth Wilmer
{"title":"数图和锦标赛的不可逆性","authors":"Noga Alon, Emil Powierski, Michael Savery, Alex Scott, Elizabeth Wilmer","doi":"10.1137/23m1547135","DOIUrl":null,"url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 327-347, March 2024. <br/> Abstract. For an oriented graph [math] and a set [math], the inversion of [math] in [math] is the digraph obtained by reversing the orientations of the edges of [math] with both endpoints in [math]. The inversion number of [math], [math], is the minimum number of inversions which can be applied in turn to [math] to produce an acyclic digraph. Answering a recent question of Bang-Jensen, da Silva, and Havet we show that, for each [math] and tournament [math], the problem of deciding whether [math] is solvable in time [math], which is tight for all [math]. In particular, the problem is fixed-parameter tractable when parameterized by [math]. On the other hand, we build on their work to prove their conjecture that for [math] the problem of deciding whether a general oriented graph [math] has [math] is NP-complete. We also construct oriented graphs with inversion number equal to twice their cycle transversal number, confirming another conjecture of Bang-Jensen, da Silva, and Havet, and we provide a counterexample to their conjecture concerning the inversion number of so-called dijoin digraphs while proving that it holds in certain cases. Finally, we asymptotically solve the natural extremal question in this setting, improving on previous bounds of Belkhechine, Bouaziz, Boudabbous, and Pouzet to show that the maximum inversion number of an [math]-vertex tournament is [math].","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"29 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2024-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Invertibility of Digraphs and Tournaments\",\"authors\":\"Noga Alon, Emil Powierski, Michael Savery, Alex Scott, Elizabeth Wilmer\",\"doi\":\"10.1137/23m1547135\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 327-347, March 2024. <br/> Abstract. For an oriented graph [math] and a set [math], the inversion of [math] in [math] is the digraph obtained by reversing the orientations of the edges of [math] with both endpoints in [math]. The inversion number of [math], [math], is the minimum number of inversions which can be applied in turn to [math] to produce an acyclic digraph. Answering a recent question of Bang-Jensen, da Silva, and Havet we show that, for each [math] and tournament [math], the problem of deciding whether [math] is solvable in time [math], which is tight for all [math]. In particular, the problem is fixed-parameter tractable when parameterized by [math]. On the other hand, we build on their work to prove their conjecture that for [math] the problem of deciding whether a general oriented graph [math] has [math] is NP-complete. We also construct oriented graphs with inversion number equal to twice their cycle transversal number, confirming another conjecture of Bang-Jensen, da Silva, and Havet, and we provide a counterexample to their conjecture concerning the inversion number of so-called dijoin digraphs while proving that it holds in certain cases. Finally, we asymptotically solve the natural extremal question in this setting, improving on previous bounds of Belkhechine, Bouaziz, Boudabbous, and Pouzet to show that the maximum inversion number of an [math]-vertex tournament is [math].\",\"PeriodicalId\":49530,\"journal\":{\"name\":\"SIAM Journal on Discrete Mathematics\",\"volume\":\"29 1\",\"pages\":\"\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-01-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"SIAM Journal on Discrete Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1137/23m1547135\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM Journal on Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1137/23m1547135","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 327-347, March 2024. Abstract. For an oriented graph [math] and a set [math], the inversion of [math] in [math] is the digraph obtained by reversing the orientations of the edges of [math] with both endpoints in [math]. The inversion number of [math], [math], is the minimum number of inversions which can be applied in turn to [math] to produce an acyclic digraph. Answering a recent question of Bang-Jensen, da Silva, and Havet we show that, for each [math] and tournament [math], the problem of deciding whether [math] is solvable in time [math], which is tight for all [math]. In particular, the problem is fixed-parameter tractable when parameterized by [math]. On the other hand, we build on their work to prove their conjecture that for [math] the problem of deciding whether a general oriented graph [math] has [math] is NP-complete. We also construct oriented graphs with inversion number equal to twice their cycle transversal number, confirming another conjecture of Bang-Jensen, da Silva, and Havet, and we provide a counterexample to their conjecture concerning the inversion number of so-called dijoin digraphs while proving that it holds in certain cases. Finally, we asymptotically solve the natural extremal question in this setting, improving on previous bounds of Belkhechine, Bouaziz, Boudabbous, and Pouzet to show that the maximum inversion number of an [math]-vertex tournament is [math].
期刊介绍:
SIAM Journal on Discrete Mathematics (SIDMA) publishes research papers of exceptional quality in pure and applied discrete mathematics, broadly interpreted. The journal''s focus is primarily theoretical rather than empirical, but the editors welcome papers that evolve from or have potential application to real-world problems. Submissions must be clearly written and make a significant contribution.
Topics include but are not limited to:
properties of and extremal problems for discrete structures
combinatorial optimization, including approximation algorithms
algebraic and enumerative combinatorics
coding and information theory
additive, analytic combinatorics and number theory
combinatorial matrix theory and spectral graph theory
design and analysis of algorithms for discrete structures
discrete problems in computational complexity
discrete and computational geometry
discrete methods in computational biology, and bioinformatics
probabilistic methods and randomized algorithms.