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.
- [1] . 2021. Flip distances between graph orientations. Algorithmica 83, 1 (2021), 116–143. Google ScholarDigital Library
- [2] Alex R. Berg and Tibor Jordán. 2006. Two-connected orientations of Eulerian graphs. J. Graph Theory 52, 3 (2006), 230–242. Google ScholarCross Ref
- [3] . 2008. Recent results on well-balanced orientations. Discr. Optim. 5, 4 (2008), 663–676. Google ScholarDigital Library
- [4] Joseph Cheriyan, Olivier Durand de Gevigney, and Zoltán Szigeti. 2014. Packing of rigid spanning subgraphs and spanning trees. J. Combin. Theory Ser. B 105 (2014), 17–25. Google ScholarDigital Library
- [5] Olivier Durand de Gevigney. 2020. On Frank’s conjecture on \(k\)-connected orientations. J. Combin. Theory Ser. B 141 (2020), 105–114. Google ScholarDigital Library
- [6] . 1982. An algorithm for submodular functions on graphs. In Bonn Workshop on Combinatorial Optimization, , , and (Eds.).
North-Holland Mathematics Studies , Vol. 66. North-Holland, 97–120. Google ScholarCross Ref - [7] . 1982. A note on \(k\)-strongly connected orientations of an undirected graph. Discr. Math. 39, 1 (1982), 103–104. Google ScholarDigital Library
- [8] . 1993. Applications of submodular functions. In Surveys in Combinatorics, 1993, (Ed.). Cambridge University Press, 85–136. Google ScholarCross Ref
- [9] . 1996. Connectivity and network flows. In Handbook of Combinatorics. Vol. 1. MIT Press, 111–177. Google Scholar
- [10] András Frank. 2011. Connections in Combinatorial Optimization. Oxford University Press. Google Scholar
- [11] András Frank, Tamás Király, and Zoltán Király. 2003. On the orientation of graphs and hypergraphs. Discr. Appl. Math. 131, 2 (2003), 385–400. Google ScholarDigital Library
- [12] . 2001. Notes on acyclic orientations and the shelling lemma. Theor. Comput. Sci. 263, 1–2 (2001), 9–16. Google ScholarDigital Library
- [13] . 2012. Graph orientations with set connectivity requirements. Discr. Math. 312, 15 (2012), 2349–2355. Google ScholarCross Ref
- [14] . 1994. Efficient splitting off algorithms for graphs. In Proceedings of the 26th Annual ACM Symposium on Theory of Computing, and (Eds.). ACM, 696–705. Google ScholarDigital Library
- [15] . 1983. On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non-Radon partitions, and orientations of graphs. Trans. Am. Math. Soc. 280 (1983), 97–126. Google ScholarCross Ref
- [16] . 1965. On the degrees of the vertices of a directed graph. J. Frankl. Inst. 279, 4 (1965), 290–308.Google ScholarCross Ref
- [17] . 2011. Dynamic lane reversal in traffic management. In Proceedings of the 14th International IEEE Conference on Intelligent Transportation Systems (ITSC’11). IEEE, 1929–1934. Google ScholarCross Ref
- [18] . 1973. Efficient algorithms for graph manipulation (algorithm 447). Commun. ACM 16, 6 (1973), 372–378. Google ScholarDigital Library
- [19] . 2022. Monotone edge flips to an orientation of maximum edge-connectivity à la Nash-Williams. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’22). 1342–1355. Google ScholarCross Ref
- [20] . 2022. Shortest reconfiguration of perfect matchings via alternating cycles. SIAM J. Discr. Math. 36, 2 (2022), 1102–1123. Google ScholarDigital Library
- [21] . 2010. An algorithm for minimum cost arc-connectivity orientations. Algorithmica 56, 4 (2010), 437–447. Google ScholarCross Ref
- [22] . 2005. On the existence of \(k\) edge-disjoint 2-connected spanning subgraphs. J. Combin. Theory Ser. B 95, 2 (2005), 257–262. Google ScholarDigital Library
- [23] . 2006. Simultaneous well-balanced orientations of graphs. J. Combin. Theory Ser. B 96, 5 (2006), 684–692. Google ScholarDigital Library
- [24] . 2013. Shattering, graph orientations, and connectivity. Electr. J. Combin. 20, P44, 3 (2013), 1–18.Google Scholar
- [25] . 1993. Combinatorial Problems and Exercises (2nd ed.). North-Holland. Google Scholar
- [26] . 1978. A minimax theorem for directed graphs. J. Lond. Math. Soc. s2–17, 3 (1978), 369–374. Google ScholarCross Ref
- [27] . 1997. Deterministic \(\tilde{O}\)\((nm)\) time edge-splitting in undirected graphs. J. Combin. Optim. 1, 1 (1997), 5–46. Google ScholarCross Ref
- [28] . 1960. On orientations, connectivity and odd-vertex-pairings in finite graphs. Can. J. Math. 12 (1960), 555–567. Google ScholarCross Ref
- [29] . 1939. A theorem on graphs, with an application to a problem of traffic control. Am. Math. Month. 46, 5 (1939), 281–283. Google ScholarCross Ref
- [30] . 1989. Configurations in graphs of large minimum degree, connectivity, or chromatic number. Ann. N. Y. Acad. Sci. 555, 1 (1989), 402–412. arXiv: https://nyaspubs.onlinelibrary.wiley.com/doi/pdf/10.1111/j.1749-6632.1989.tb22479.x.Google ScholarCross Ref
- [31] . 2015. Strongly 2-connected orientations of graphs. J. Combin. Theory Ser. B 110 (2015), 67–78. Google ScholarDigital Library
Index Terms
- Monotone Edge Flips to an Orientation of Maximum Edge-Connectivity à la Nash-Williams
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