Invertibility of Digraphs and Tournaments

IF 0.9 3区 数学 Q2 MATHEMATICS
Noga Alon, Emil Powierski, Michael Savery, Alex Scott, Elizabeth Wilmer
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引用次数: 0

Abstract

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].
数图和锦标赛的不可逆性
SIAM 离散数学杂志》,第 38 卷,第 1 期,第 327-347 页,2024 年 3 月。 摘要。对于一个有向图[math]和一个集合[math],[math]在[math]中的反转是将[math]中两个端点都在[math]中的边的方向反转而得到的数图。数学]的反转数[math]是依次应用于[数学]以产生非循环数图的最小反转数。为了回答邦-简森(Bang-Jensen)、达-席尔瓦(da Silva)和哈特(Havet)最近提出的一个问题,我们证明,对于每一个[数学]和锦标赛[数学],决定[数学]是否可解的问题在时间[数学]内是可解的,这对所有[数学]来说都是紧的。特别是,当以[math]为参数时,该问题是固定参数可解的。另一方面,我们以他们的工作为基础,证明了他们的猜想,即对于 [math],判断一般面向图 [math] 是否具有 [math] 的问题是 NP-完全的。我们还构造了反转数等于其循环横向数两倍的定向图,证实了 Bang-Jensen、da Silva 和 Havet 的另一个猜想,并提供了他们关于所谓二重连接图反转数猜想的反例,同时证明该猜想在某些情况下成立。最后,我们渐近地解决了在这种情况下的自然极值问题,改进了贝尔赫钦、布阿齐兹、布达布斯和普泽特之前的界限,证明了[数学]顶点锦标赛的最大反转数是[数学]。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
1.90
自引率
0.00%
发文量
124
审稿时长
4-8 weeks
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.
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