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Edge-Orders

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Abstract

Canonical orderings and their relatives such as st-numberings have been used as a key tool in algorithmic graph theory for the last decades. Recently, a unifying link behind all these orders has been shown that links them to well-known graph decompositions into parts that have a prescribed vertex-connectivity. Despite extensive interest in canonical orderings, no analogue of this unifying concept is known for edge-connectivity. In this paper, we establish such a concept named edge-orders and show how to compute (1, 1)-edge-orders of 2-edge-connected graphs as well as (2, 1)-edge-orders of 3-edge-connected graphs in linear time, respectively. While the former can be seen as the edge-variants of st-numberings, the latter are the edge-variants of Mondshein sequences and non-separating ear decompositions. The methods that we use for obtaining such edge-orders differ considerably in almost all details from the ones used for their vertex-counterparts, as different graph-theoretic constructions are used in the inductive proof and standard reductions from edge- to vertex-connectivity are bound to fail. As a first application, we consider the famous Edge-Independent Spanning Tree Conjecture, which asserts that every k-edge-connected graph contains k rooted spanning trees that are pairwise edge-independent. We illustrate the impact of the above edge-orders by deducing algorithms that construct 2- and 3-edge independent spanning trees of 2- and 3-edge-connected graphs, the latter of which improves the best known running time from \(O(n^2)\) to linear time.

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References

  1. Annexstein, F., Berman, K., Swaminathan, R.: Independent spanning trees with small stretch factors. Technical Report 96-13, DIMACS (June 1996)

  2. Badent, M., Brandes, U., Cornelsen, S.: More canonical ordering. J. Graph Algorithms Appl. 15(1), 97–126 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bender, M.A., Cole, R., Demaine, E.D., Farach-Colton, M., Zito, J.: Two simplified algorithms for maintaining order in a list. In: Proceedings of the 10th European Symposium on Algorithms (ESA’02), pp. 152–164 (2002)

  4. Biedl, T., Derka, M.: The (3,1)-ordering for 4-connected planar triangulations. J. Graph Algorithms Appl. 20(2), 347–362 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Biedl, T., Schmidt, J.M.: Small-area orthogonal drawings of 3-connected graphs. In: Proceedings of the 23rd International Symposium on Graph Drawing (GD’15), pp. 153–165 (2015)

  6. Cheriyan, J., Maheshwari, S.N.: Finding nonseparating induced cycles and independent spanning trees in 3-connected graphs. J. Algorithms 9(4), 507–537 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  7. Curran, S., Lee, O., Yu, X.: Chain decompositions of 4-connected graphs. SIAM J. Discrete Math. 19(4), 848–880 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. de Fraysseix, H., Pach, J., Pollack, R.: Small sets supporting fary embeddings of planar graphs. In: Proceedings of the 20th Annual ACM Symposium on Theory of Computing (STOC’88), pp. 426–433 (1988)

  9. Djidjev, H.N.: A linear-time algorithm for finding a maximal planar subgraph. SIAM J. Discrete Math. 20(2), 444–462 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Even, S., Tarjan, R.E.: Computing an st-Numbering. Theor. Comput. Sci. 2(3), 339–344 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gabow, H.N., Tarjan, R.E.: A linear-time algorithm for a special case of disjoint set union. J. Comput. Syst. Sci. 30(2), 209–221 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  12. Galil, Z., Italiano, G.F.: Reducing edge connectivity to vertex connectivity. SIGACT News 22(1), 57–61 (1991)

    Article  Google Scholar 

  13. Gopalan, A., Ramasubramanian, S.: A counterexample for the proof of implication conjecture on independent spanning trees. Inf. Process. Lett. 113(14–16), 522–526 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gopalan, A., Ramasubramanian, S.: IP fast rerouting and disjoint multipath routing with three edge-independent spanning trees. IEEE/ACM Trans. Netw. 24(3), 1336–1349 (2016)

    Article  Google Scholar 

  15. Hoyer, A., Thomas, R.: Four edge-independent spanning trees. SIAM J. Discrete Math. 32(1), 233–248 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  16. Imai, H., Asano, T.: Dynamic orthogonal segment intersection search. J. Algorithms 8(1), 1–18 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  17. Itai, A., Rodeh, M.: The multi-tree approach to reliability in distributed networks. In: 25th Annual Symposium on Foundations of Computer Science (FOCS’84), pp. 137–147 (1984)

  18. Kant, G.: Drawing planar graphs using the lmc-ordering. In: Proceedings of the 33th Annual Symposium on Foundations of Computer Science (FOCS’92), pp. 101–110 (1992)

  19. Lovász, L.: Computing ears and branchings in parallel. In: Proceedings of the 26th Annual Symposium on Foundations of Computer Science (FOCS’85), pp. 464–467 (1985)

  20. Mader, W.: A reduction method for edge-connectivity in graphs. In: Bollobás, B. (ed.) Advances in Graph Theory. Annals of Discrete Mathematics, vol. 3, pp. 145–164. North-Holland, Amsterdam (1978)

    Chapter  Google Scholar 

  21. McConnell, R.M., Mehlhorn, K., Näher, S., Schweitzer, P.: Certifying algorithms. Comput. Sci. Rev. 5(2), 119–161 (2011)

    Article  MATH  Google Scholar 

  22. Mehlhorn, K., Neumann, A., Schmidt, J.M.: Certifying 3-edge-connectivity. Algorithmica 77(2), 309–335 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mondshein, L.F.: Combinatorial Ordering and the Geometric Embedding of Graphs. Ph.D. thesis, M.I.T. Lincoln Laboratory/Harvard University (1971). Technical Report available at www.dtic.mil/cgi-bin/GetTRDoc?AD=AD0732882. Accessed 11 Sept 2018

  24. Nagai, S., Nakano, S.: A linear-time algorithm to find independent spanning trees in maximal planar graphs. In: 26th International Workshop on Graph-Theoretic Concepts in Computer Science (WG’00), pp. 290–301 (2000)

  25. Nakano, S., Rahman, M.S., Nishizeki, T.: A linear-time algorithm for four-partitioning four-connected planar graphs. Inf. Process. Lett. 62(6), 315–322 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  26. Robbins, H.E.: A theorem on graphs, with an application to a problem of traffic control. Am. Math. Mon. 46(5), 281–283 (1939)

    Article  MATH  Google Scholar 

  27. Schmidt, J.M.: Construction sequences and certifying 3-connectedness. In: Proceedings of the 27th Symposium on Theoretical Aspects of Computer Science (STACS’10), pp. 633–644 (2010)

  28. Schmidt, J.M.: A simple test on 2-vertex- and 2-edge-connectivity. Inf. Process. Lett. 113(7), 241–244 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. Schmidt, J.M.: The Mondshein sequence. In: Proceedings of the 41st International Colloquium on Automata, Languages and Programming (ICALP’14), pp. 967–978 (2014)

  30. Schmidt, J.M.: Mondshein sequences (a.k.a. (2,1)-orders). SIAM J. Comput. 45(6), 1985–2003 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. Whitney, H.: Non-separable and planar graphs. Trans. Am. Math. Soc. 34(1), 339–362 (1932)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Lena Schlipf.

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An extended abstract of this paper appeared at ICALP’17.

J. M. Schmidt: This research is supported by the Grant SCHM 3186/1-1 (270450205) from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation).

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Schlipf, L., Schmidt, J.M. Edge-Orders. Algorithmica 81, 1881–1900 (2019). https://doi.org/10.1007/s00453-018-0516-4

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