Pathwidth Versus Cocircumference

IF 0.9 3区数学 Q2 MATHEMATICS

SIAM Journal on Discrete Mathematics Pub Date : 2024-02-26 DOI:10.1137/23m158663x

Marcin Briański, Gwenaël Joret, Michał T. Seweryn

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引用次数: 0

Abstract

SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 857-866, March 2024.
Abstract. The circumference of a graph [math] with at least one cycle is the length of a longest cycle in [math]. A classic result of Birmelé [J. Graph Theory, 43 (2003), pp. 24–25] states that the treewidth of [math] is at most its circumference minus 1. In case [math] is 2-connected, this upper bound also holds for the pathwidth of [math]; in fact, even the treedepth of [math] is upper bounded by its circumference (Briański et al. [Treedepth vs circumference, Combinatorica, 43 (2023), pp. 659–664]). In this paper, we study whether similar bounds hold when replacing the circumference of [math] by its cocircumference, defined as the largest size of a bond in [math], an inclusionwise minimal set of edges [math] such that [math] has more components than [math]. In matroidal terms, the cocircumference of [math] is the circumference of the bond matroid of [math]. Our first result is the following “dual” version of Birmelé’s theorem: The treewidth of a graph [math] is at most its cocircumference. Our second and main result is an upper bound of [math] on the pathwidth of a 2-connected graph [math] with cocircumference [math]. Contrary to circumference, no such bound holds for the treedepth of [math]. Our two upper bounds are best possible up to a constant factor.

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路径宽度与头围

SIAM 离散数学杂志》，第 38 卷第 1 期，第 857-866 页，2024 年 3 月。摘要。至少有一个循环的图[math]的周长是[math]中最长循环的长度。Birmelé [J. Graph Theory, 43 (2003), pp. 24-25] 的一个经典结果指出，[math] 的树宽最多为周长减 1。如果[math]是 2 连接的，那么[math]的路径宽度也有这个上界；事实上，甚至[math]的树深也有其周长的上界（Briański 等人 [Treedepth vs circumference, Combinatorica, 43 (2023), pp. 659-664]）。在本文中，我们将研究当用[math]的cocircumference 代替[math]的周长时，类似的边界是否成立。cocircumference 的定义是[math]中一个结合点的最大尺寸，它是边[math]的包容最小集，使得[math]的成分多于[math]。用母陀螺术语来说，[math] 的cocircumference 就是[math] 的bond matroid 的周长。我们的第一个结果是以下伯梅莱定理的 "对偶 "版本：一个图[math]的树宽最多是它的cocircumference。我们的第二个也是主要的结果是[math]关于具有圆周率[math]的 2 连接图[math]的路径宽度的上界[math]。与圆周率相反，[math] 的树深度不存在这样的上界。我们的两个上界在一个常数因子以内都是最好的。

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

SIAM Journal on Discrete Mathematics 数学-应用数学

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.