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Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)

Monotone edge flips to an orientation of maximum edge-connectivity à la Nash-Williams

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: an 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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cover image Proceedings
Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)
Pages: 1342 - 1355
Editors: Joseph (Seffi) Naor, Technion Israel Institute of Technology, Israel and Niv Buchbinder, Tel Aviv University, Israel
ISBN (Online): 978-1-61197-707-3

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Published online: 5 January 2022

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The full version of the paper can be accessed at https://arxiv.org/abs/2110.11585

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