Monotone Edge Flips to an Orientation of Maximum Edge-Connectivity à la Nash-Williams

IF 0.9 3区计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS

ACM Transactions on Algorithms Pub Date : 2021-10-22 DOI:10.1145/3561302

Takehiro Ito, Yuni Iwamasa, Naonori Kakimura, Naoyuki Kamiyama, Yusuke Kobayashi, Shun-ichi Maezawa, Yuta Nozaki, Y. Okamoto, K. Ozeki

引用次数: 5

Abstract

We initiate the study of k-edge-connected orientations of undirected graphs through edge flips for k ≥ 2. We prove that in every orientation of an undirected 2k-edge-connected graph, there exists a sequence of edges such that flipping their directions one by one does not decrease the edge connectivity, and the final orientation is k-edge connected. This yields an “edge-flip based” new proof of Nash-Williams’ theorem: A undirected graph G has a k-edge-connected orientation if and only if G is 2k-edge connected. As another consequence of the theorem, we prove that the edge-flip graph of k-edge-connected orientations of an undirected graph G is connected if G is (2k+2)-edge connected. This has been known to be true only when k=1.

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单调边向Nash-Williams最大边连通性的方向翻转

通过k≥2的边翻转，我们开始研究无向图的k边连通方向。我们证明了在无向2k边连通图的每个方向上，存在一个边序列，使得它们的方向一个接一个地翻转不会降低边连通性，并且最终的方向是k-边连通的。这给出了Nash-Williams定理的一个“基于边翻转”的新证明：无向图G具有k-边连通方向，当且仅当G是2k边连通的。作为定理的另一个结果，我们证明了无向图G的k-边连通方向的边翻转图是连通的，如果G是（2k+2）-边连通的。已知只有当k＝1时，这才是真的。

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

ACM Transactions on Algorithms COMPUTER SCIENCE, THEORY & METHODS-MATHEMATICS, APPLIED

CiteScore

3.30

自引率

0.00%

发文量

审稿时长

6-12 weeks

期刊介绍： ACM Transactions on Algorithms welcomes submissions of original research of the highest quality dealing with algorithms that are inherently discrete and finite, and having mathematical content in a natural way, either in the objective or in the analysis. Most welcome are new algorithms and data structures, new and improved analyses, and complexity results. Specific areas of computation covered by the journal include combinatorial searches and objects; counting; discrete optimization and approximation; randomization and quantum computation; parallel and distributed computation; algorithms for graphs, geometry, arithmetic, number theory, strings; on-line analysis; cryptography; coding; data compression; learning algorithms; methods of algorithmic analysis; discrete algorithms for application areas such as biology, economics, game theory, communication, computer systems and architecture, hardware design, scientific computing