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A Pivot Gray Code Listing for the Spanning Trees of the Fan Graph

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Computing and Combinatorics (COCOON 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 13025))

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Abstract

We use a greedy strategy to list the spanning trees of the fan graph, \(F_n\), such that successive trees differ by pivoting a single edge around a vertex. It is the first greedy algorithm for exhaustively generating spanning trees using such a minimal change operation. The resulting listing is then studied to find a recursive algorithm that produces the same listing in O(1)-amortized time using O(n) space. Additionally, we present O(n)-time algorithms for ranking and unranking the spanning trees for our listing; an improvement over the generic \(O(n^3)\)-time algorithm for ranking and unranking spanning trees of an arbitrary graph.

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References

  1. Berger, I.: The enumeration of trees without duplication. IEEE Trans. Circuit Theory 14(4), 417–418 (1967). https://doi.org/10.1109/TCT.1967.1082758

    Article  Google Scholar 

  2. Bogdanowicz, Z.R.: Formulas for the number of spanning trees in a fan. Appl. Math. Sci. 2(16), 781–786 (2008)

    MathSciNet  MATH  Google Scholar 

  3. Chakraborty, M., Chowdhury, S., Chakraborty, J., Mehera, R., Pal, R.K.: Algorithms for generating all possible spanning trees of a simple undirected connected graph: an extensive review. Complex Intell. Syst. 5(3), 265–281 (2018). https://doi.org/10.1007/s40747-018-0079-7

    Article  Google Scholar 

  4. Char, J.: Generation of trees, two-trees, and storage of master forests. IEEE Trans. Circuit Theory 15(3), 228–238 (1968)

    Article  MathSciNet  Google Scholar 

  5. Colbourn, C.J., Day, R.P., Nel, L.D.: Unranking and ranking spanning trees of a graph. J. Algorithms 10(2), 271–286 (1989)

    Article  MathSciNet  Google Scholar 

  6. Cummins, R.: Hamilton circuits in tree graphs. IEEE Trans. Circuit Theory 13(1), 82–90 (1966). https://doi.org/10.1109/TCT.1966.1082546

    Article  MathSciNet  Google Scholar 

  7. Feussner, W.: Ueber stromverzweigung in netzförmigen leitern. Ann. Phys. 314(13), 1304–1329 (1902)

    Article  Google Scholar 

  8. Gabow, H.N., Myers, E.W.: Finding all spanning trees of directed and undirected graphs. SIAM J. Comput. 7(3), 280–287 (1978)

    Article  MathSciNet  Google Scholar 

  9. Hakimi, S.: On trees of a graph and their generation. J. Franklin Inst. 272(5), 347–359 (1961)

    Article  MathSciNet  Google Scholar 

  10. Hartung, E., Hoang, H.P., Mütze, T., Williams, A.: Combinatorial generation via permutation languages. In: Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1214–1225. SIAM (2020)

    Google Scholar 

  11. Hoang, H.P., Mütze, T.: Combinatorial generation via permutation languages. II. Lattice congruences. arXiv preprint arXiv:1911.12078 (2019)

  12. Holzmann, C.A., Harary, F.: On the tree graph of a matroid. SIAM J. Appl. Math. 22(2), 187–193 (1972). https://doi.org/10.1137/0122021

    Article  MathSciNet  MATH  Google Scholar 

  13. Kamae, T.: The existence of a Hamilton circuit in a tree graph. IEEE Trans. Circuit Theory 14(3), 279–283 (1967)

    Article  MathSciNet  Google Scholar 

  14. Kapoor, S., Ramesh, H.: Algorithms for enumerating all spanning trees of undirected and weighted graphs. SIAM J. Comput. 24(2), 247–265 (1995)

    Article  MathSciNet  Google Scholar 

  15. Kishi, G., Kajitani, Y.: On Hamilton circuits in tree graphs. IEEE Trans. Circuit Theory 15(1), 42–50 (1968). https://doi.org/10.1109/TCT.1968.1082762

    Article  Google Scholar 

  16. Matsui, T.: A flexible algorithm for generating all the spanning trees in undirected graphs. Algorithmica 18, 530–543 (1997)

    Article  MathSciNet  Google Scholar 

  17. Mayeda, W., Seshu, S.: Generation of trees without duplications. IEEE Trans. Circuit Theory 12(2), 181–185 (1965)

    Article  MathSciNet  Google Scholar 

  18. Merino, A., Mütze, T.: Efficient generation of rectangulations via permutation languages. In: 37th International Symposium on Computational Geometry (SoCG 2021). Schloss Dagstuhl-Leibniz-Zentrum für Informatik (2021)

    Google Scholar 

  19. Minty, G.: A simple algorithm for listing all the trees of a graph. IEEE Trans. Circuit Theory 12(1), 120 (1965)

    Article  Google Scholar 

  20. Shioura, A., Tamura, A.: Efficiently scanning all spanning trees of an undirected graph. J. Oper. Res. Soc. Jpn. 38(3), 331–344 (1995)

    MathSciNet  MATH  Google Scholar 

  21. Shioura, A., Tamura, A., Uno, T.: An optimal algorithm for scanning all spanning trees of undirected graphs. SIAM J. Comput. 26(3), 678–692 (1997)

    Article  MathSciNet  Google Scholar 

  22. Smith, M.J.: Generating spanning trees. Master’s thesis, University of Victoria (1997)

    Google Scholar 

  23. Williams, A.: The greedy gray code algorithm. In: Dehne, F., Solis-Oba, R., Sack, J.-R. (eds.) WADS 2013. LNCS, vol. 8037, pp. 525–536. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40104-6_46

    Chapter  Google Scholar 

  24. Winter, P.: An algorithm for the enumeration of spanning trees. BIT Numer. Math. 26, 44–62 (1985)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. University of Guelph, Guelph, Canada

    Ben Cameron, Aaron Grubb & Joe Sawada

Authors
  1. Ben Cameron
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  2. Aaron Grubb
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  3. Joe Sawada
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Corresponding author

Correspondence to Aaron Grubb .

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Editors and Affiliations

  1. National Cheng Kung University, Tainan, Taiwan

    Chi-Yeh Chen

  2. National Tsing Hua University, Hsinchu, Taiwan

    Wing-Kai Hon

  3. National Taipei University of Business, Taoyuan, Taiwan

    Ling-Ju Hung

  4. National Taitung University, Taitung, Taiwan

    Chia-Wei Lee

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Cameron, B., Grubb, A., Sawada, J. (2021). A Pivot Gray Code Listing for the Spanning Trees of the Fan Graph. In: Chen, CY., Hon, WK., Hung, LJ., Lee, CW. (eds) Computing and Combinatorics. COCOON 2021. Lecture Notes in Computer Science(), vol 13025. Springer, Cham. https://doi.org/10.1007/978-3-030-89543-3_5

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  • DOI: https://doi.org/10.1007/978-3-030-89543-3_5

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-89542-6

  • Online ISBN: 978-3-030-89543-3

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