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Listing Spanning Trees of Outerplanar Graphs by Pivot-Exchanges

Authors Nastaran Behrooznia, Torsten Mütze

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Author Details

Nastaran Behrooznia
  • Department of Computer Science, University of Warwick, Coventry, UK
Torsten Mütze
  • Institut für Mathematik, Universität Kassel, Germany
  • Department of Theoretical Computer Science and Mathematical Logic, Charles University, Prague, Czech Republic

Funding

This work was supported by Czech Science Foundation grant GA 22-15272S. Both authors participated in the workshop "Combinatorics, Algorithms and Geometry" in March 2024, which was funded by German Science Foundation grant 522790373.

Acknowledgements

We thank both reviewers of this paper for a number of very helpful suggestions that improved the writing.

Cite As Get BibTex

Nastaran Behrooznia and Torsten Mütze. Listing Spanning Trees of Outerplanar Graphs by Pivot-Exchanges. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 16:1-16:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.STACS.2025.16

Abstract

We prove that the spanning trees of any outerplanar triangulation G can be listed so that any two consecutive spanning trees differ in an exchange of two edges that share an end vertex. For outerplanar graphs G with faces of arbitrary lengths (not necessarily 3) we establish a similar result, with the condition that the two exchanged edges share an end vertex or lie on a common face. These listings of spanning trees are obtained from a simple greedy algorithm that can be implemented efficiently, i.e., in time {O}(n log n) per generated spanning tree, where n is the number of vertices of G. Furthermore, the listings correspond to Hamilton paths on the 0/1-polytope that is obtained as the convex hull of the characteristic vectors of all spanning trees of G.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Trees
Keywords
  • Spanning tree
  • generation
  • edge exchange
  • Hamilton path
  • Gray code

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References

  1. D. J. Aldous. The random walk construction of uniform spanning trees and uniform labelled trees. SIAM J. Discrete Math., 3(4):450-465, 1990. URL: https://doi.org/10.1137/0403039.
  2. A. Z. Broder. Generating random spanning trees. In 30th Annual Symposium on Foundations of Computer Science, Research Triangle Park, North Carolina, USA, 30 October - 1 November 1989, pages 442-447. IEEE Computer Society, 1989. URL: https://doi.org/10.1109/SFCS.1989.63516.
  3. B. Cameron, A. Grubb, and J. Sawada. Pivot Gray codes for the spanning trees of a graph ft. the fan. Graphs Combin., 40(4):Paper No. 78, 26 pp., 2024. URL: https://doi.org/10.1007/s00373-024-02808-2.
  4. M. Chakraborty, S. Chowdhury, J. Chakraborty, R. Mehera, and R. K. Pal. Algorithms for generating all possible spanning trees of a simple undirected connected graph: an extensive review. Complex & Intelligent Systems, 5:265-281, 2019. Google Scholar
  5. W.-K. Chen. Hamilton circuits in directed-tree graphs. IEEE Trans. Circuit Theory, CT-14:231-233, 1967. Google Scholar
  6. C. J. Colbourn, R. P. J. Day, and L. D. Nel. Unranking and ranking spanning trees of a graph. J. Algorithms, 10(2):271-286, 1989. URL: https://doi.org/10.1016/0196-6774(89)90016-3.
  7. C. J. Colbourn, W. J. Myrvold, and E. Neufeld. Two algorithms for unranking arborescences. J. Algorithms, 20(2):268-281, 1996. URL: https://doi.org/10.1006/jagm.1996.0014.
  8. R. L. Cummins. Hamilton circuits in tree graphs. IEEE Trans. Circuit Theory, CT-13:82-90, 1966. Google Scholar
  9. H. N. Gabow and E. W. Myers. Finding all spanning trees of directed and undirected graphs. SIAM J. Comput., 7(3):280-287, 1978. URL: https://doi.org/10.1137/0207024.
  10. C. A. Holzmann and F. Harary. On the tree graph of a matroid. SIAM J. Appl. Math., 22:187-193, 1972. URL: https://doi.org/10.1137/0122021.
  11. T. Kamae. The existence of a Hamilton circuit in a tree graph. IEEE Trans. Circuit Theory, CT-14:279-283, 1967. Google Scholar
  12. S. Kapoor and H. Ramesh. Algorithms for generating all spanning trees of undirected, directed and weighted graphs. In Algorithms and data structures (Ottawa, ON, 1991), volume 519 of Lecture Notes in Comput. Sci., pages 461-472. Springer, Berlin, 1991. URL: https://doi.org/10.1007/BFb0028284.
  13. G. Kishi and Y. Kajitani. On Hamilton circuits in tree graphs. IEEE Trans. Circuit Theory, CT-15(1):42-50, 1968. Google Scholar
  14. D. E. Knuth. The Art of Computer Programming. Vol. 4A. Combinatorial Algorithms. Part 1. Addison-Wesley, Upper Saddle River, NJ, 2011. Google Scholar
  15. T. Matsui. A flexible algorithm for generating all the spanning trees in undirected graphs. Algorithmica, 18(4):530-543, 1997. URL: https://doi.org/10.1007/PL00009171.
  16. A. Merino and T. Mütze. Traversing combinatorial 0/1-polytopes via optimization. SIAM J. Comput., 53(5):1257-1292, 2024. URL: https://doi.org/10.1137/23M1612019.
  17. A. Merino, T. Mütze, and A. Williams. All your base ̲ s are belong to us: listing all bases of a matroid by greedy exchanges. In 11th International Conference on Fun with Algorithms, volume 226 of LIPIcs. Leibniz Int. Proc. Inform., pages Paper No. 22, 28. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/lipics.fun.2022.22.
  18. G. Minty. A simple algorithm for listing all the trees of a graph. IEEE Trans. Circuit Theory, 12(1):120, 1965. Google Scholar
  19. T. Mütze. Combinatorial Gray codes - an updated survey. Electron. J. Combin., DS26(Dynamic Surveys):99 pp., 2023. URL: https://doi.org/10.37236/11023.
  20. D. J. Naddef and W. R. Pulleyblank. Hamiltonicity and combinatorial polyhedra. J. Combin. Theory Ser. B, 31(3):297-312, 1981. URL: https://doi.org/10.1016/0095-8956(81)90032-0.
  21. D. J. Naddef and W. R. Pulleyblank. Hamiltonicity in (0-1)-polyhedra. J. Combin. Theory Ser. B, 37(1):41-52, 1984. URL: https://doi.org/10.1016/0095-8956(84)90043-1.
  22. V. Rao and N. Raju. On tree graphs of directed graphs. IEEE Trans. Circuit Theory, 19(3):282-283, 1972. Google Scholar
  23. R. C. Read and R. E. Tarjan. Bounds on backtrack algorithms for listing cycles, paths, and spanning trees. Networks, 5(3):237-252, 1975. URL: https://doi.org/10.1002/net.1975.5.3.237.
  24. C. D. Savage. A survey of combinatorial Gray codes. SIAM Rev., 39(4):605-629, 1997. URL: https://doi.org/10.1137/S0036144595295272.
  25. A. Schrijver. Combinatorial optimization. Polyhedra and efficiency. Vol. B, volume 24 of Algorithms and Combinatorics. Springer-Verlag, Berlin, 2003. Matroids, trees, stable sets, Chapters 39-69. Google Scholar
  26. H. Shank. A note on Hamilton circuits in tree graphs. IEEE Trans. Circuit Theory, 15(1):86, 1968. URL: https://doi.org/10.1109/TCT.1968.1082765.
  27. A. Shioura and A. Tamura. Efficiently scanning all spanning trees of an undirected graph. J. Oper. Res. Soc. Japan, 38(3):331-344, 1995. URL: https://doi.org/10.15807/jorsj.38.331.
  28. A. Shioura, A. Tamura, and T. Uno. An optimal algorithm for scanning all spanning trees of undirected graphs. SIAM J. Comput., 26(3):678-692, 1997. URL: https://doi.org/10.1137/S0097539794270881.
  29. P. J. Slater. Fibonacci numbers in the count of spanning trees. Fibonacci Quart., 15(1):11-14, 1977. Google Scholar
  30. M. J. Smith. Generating spanning trees. Master’s thesis, University of Victoria, 1997. Google Scholar
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