Document Open Access Logo

The Isomorphism Problem of Power Graphs and a Question of Cameron

Authors Bireswar Das, Jinia Ghosh , Anant Kumar

PDF
Thumbnail PDF

File

LIPIcs.FSTTCS.2024.20.pdf
  • Filesize: 0.83 MB
  • 23 pages

Document Identifiers

Author Details

Bireswar Das
  • Indian Institute of Technology Gandhinagar, India
Jinia Ghosh
  • Indian Institute of Technology Gandhinagar, India
Anant Kumar
  • Indian Institute of Technology Gandhinagar, India

Cite As Get BibTex

Bireswar Das, Jinia Ghosh, and Anant Kumar. The Isomorphism Problem of Power Graphs and a Question of Cameron. In 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 323, pp. 20:1-20:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.FSTTCS.2024.20

Abstract

We study the isomorphism problem of graphs that are defined in terms of groups, namely power graphs, directed power graphs, and enhanced power graphs. We design polynomial-time algorithms for the isomorphism problems for the power graphs, the directed power graphs and the enhanced power graphs arising from finite nilpotent groups. In contrast, no polynomial-time algorithm is known for the group isomorphism problem, even for nilpotent groups of class 2. 
Our algorithms do not require the underlying groups of the input graphs to be given. A crucial step in our algorithms is to reconstruct the directed power graph from the given power graph or the enhanced power graph. The problem of efficiently computing the directed power graph from a power graph or an enhanced power graph is due to Cameron [IJGT'22]. Bubboloni and Pinzauti [Arxiv'22] gave a polynomial-time algorithm to reconstruct the directed power graph from a power graph. We give an efficient algorithm to compute the directed power graph from an enhanced power graph. The tools and techniques that we design are general enough to give a different algorithm to compute the directed power graph from a power graph as well.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Graph algorithms analysis
Keywords
  • Graph Isomorphism
  • Graphs defined on Groups
  • Power Graphs
  • Enhanced Power Graphs
  • Directed Power Graphs
  • Nilpotent Groups

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Alfred V. Aho and John E. Hopcroft. The design and analysis of computer algorithms. Pearson Education India, 1974. Google Scholar
  2. László Babai. Graph isomorphism in quasipolynomial time. In Proceedings of the forty-eighth annual ACM symposium on Theory of Computing, pages 684-697, 2016. Google Scholar
  3. László Babai, D. Yu Grigoryev, and David M. Mount. Isomorphism of graphs with bounded eigenvalue multiplicity. In Proceedings of the fourteenth annual ACM symposium on Theory of computing, pages 310-324, 1982. URL: https://doi.org/10.1145/800070.802206.
  4. László Babai and Eugene M Luks. Canonical labeling of graphs. In Proceedings of the fifteenth annual ACM symposium on Theory of computing, pages 171-183, 1983. URL: https://doi.org/10.1145/800061.808746.
  5. Hans L. Bodlaender. Polynomial algorithms for graph isomorphism and chromatic index on partial k-trees. Journal of Algorithms, 11(4):631-643, 1990. URL: https://doi.org/10.1016/0196-6774(90)90013-5.
  6. Ravi B. Boppana, Johan Hastad, and Stathis Zachos. Does co-np have short interactive proofs? Information Processing Letters, 25(2):127-132, 1987. URL: https://doi.org/10.1016/0020-0190(87)90232-8.
  7. Daniela Bubboloni and Nicolas Pinzauti. Critical classes of power graphs and reconstruction of directed power graphs. arXiv preprint arXiv:2211.14778, 2022. Google Scholar
  8. Peter J. Cameron. The power graph of a finite group, ii. Journal of Group Theory, 13(6):779-783, 2010. Google Scholar
  9. Peter J. Cameron. Graphs defined on groups. International Journal of Group Theory, 11(2):53-107, 2022. Google Scholar
  10. Peter J. Cameron and Shamik Ghosh. The power graph of a finite group. Discrete Mathematics, 311(13):1220-1222, 2011. URL: https://doi.org/10.1016/J.DISC.2010.02.011.
  11. Ivy Chakrabarty, Shamik Ghosh, and M. K. Sen. Undirected power graphs of semigroups. In Semigroup Forum, volume 78, pages 410-426. Springer, 2009. Google Scholar
  12. Bireswar Das, Jinia Ghosh, and Anant Kumar. The isomorphism problem of power graphs and a question of cameron. arXiv preprint arXiv:2305.18936, 2023. URL: https://doi.org/10.48550/arXiv.2305.18936.
  13. Min Feng, Xuanlong Ma, and Kaishun Wang. The full automorphism group of the power (di) graph of a finite group. European Journal of Combinatorics, 52:197-206, 2016. URL: https://doi.org/10.1016/J.EJC.2015.10.006.
  14. Valentina Grazian and Carmine Monetta. A conjecture related to the nilpotency of groups with isomorphic non-commuting graphs. Journal of Algebra, 633:389-402, 2023. Google Scholar
  15. Joshua A. Grochow and Youming Qiao. On p-group isomorphism: search-to-decision, counting-to-decision, and nilpotency class reductions via tensors. In 36th Computational Complexity Conference (CCC 2021), volume 200, 2021. URL: https://doi.org/10.4230/LIPICS.CCC.2021.16.
  16. Martin Grohe and Sandra Kiefer. A linear upper bound on the Weisfeiler-Leman dimension of graphs of bounded genus. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. Google Scholar
  17. Martin Grohe and Daniel Neuen. Isomorphism, canonization, and definability for graphs of bounded rank width. Communications of the ACM, 64(5):98-105, 2021. URL: https://doi.org/10.1145/3453943.
  18. Martin Grohe, Daniel Neuen, and Pascal Schweitzer. A faster isomorphism test for graphs of small degree. SIAM Journal on Computing, pages FOCS18-1, 2020. Google Scholar
  19. Martin Grohe, Daniel Neuen, Pascal Schweitzer, and Daniel Wiebking. An improved isomorphism test for bounded-tree-width graphs. ACM Transactions on Algorithms (TALG), 16(3):1-31, 2020. URL: https://doi.org/10.1145/3382082.
  20. Martin Grohe and Pascal Schweitzer. Isomorphism testing for graphs of bounded rank width. In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, pages 1010-1029. IEEE, 2015. URL: https://doi.org/10.1109/FOCS.2015.66.
  21. Marc Hellmuth and Tilen Marc. On the cartesian skeleton and the factorization of the strong product of digraphs. Theoretical Computer Science, 565:16-29, 2015. URL: https://doi.org/10.1016/J.TCS.2014.10.045.
  22. Wilfried Imrich, Sandi Klavzar, and Douglas F. Rall. Topics in graph theory: Graphs and their Cartesian product. CRC Press, 2008. Google Scholar
  23. G. James Oxley. Matroid Theory. Oxford graduate texts in mathematics. Oxford University Press, 2006. Google Scholar
  24. Andrei V. Kelarev and Stephen J. Quinn. A combinatorial property and power graphs of groups. Contributions to general algebra, 12(58):3-6, 2000. Google Scholar
  25. Ajay Kumar, Lavanya Selvaganesh, Peter J. Cameron, and T. Tamizh Chelvam. Recent developments on the power graph of finite groups-a survey. AKCE International Journal of Graphs and Combinatorics, 18(2):65-94, 2021. URL: https://doi.org/10.1080/09728600.2021.1953359.
  26. Eugene M. Luks. Isomorphism of graphs of bounded valence can be tested in polynomial time. Journal of computer and system sciences, 25(1):42-65, 1982. URL: https://doi.org/10.1016/0022-0000(82)90009-5.
  27. Ralph McKenzie. Cardinal multiplication of structures with a reflexive relation. Fundamenta Mathematicae, 70(1):59-101, 1971. Google Scholar
  28. Gary Miller. Isomorphism testing for graphs of bounded genus. In Proceedings of the twelfth annual ACM symposium on Theory of computing, pages 225-235, 1980. URL: https://doi.org/10.1145/800141.804670.
  29. Gary L. Miller. On the nlog n isomorphism technique (a preliminary report). In Proceedings of the tenth annual ACM symposium on theory of computing, pages 51-58, 1978. URL: https://doi.org/10.1145/800133.804331.
  30. Dave Witte Morris, Joy Morris, and Gabriel Verret. Isomorphisms of cayley graphs on nilpotent groups. New York J. Math, 22:453-467, 2016. Google Scholar
  31. Himadri Mukherjee. Hamiltonian cycles of power graph of abelian groups. Afrika Matematika, 30:1025-1040, 2019. Google Scholar
  32. Daniel Neuen. Isomorphism testing for graphs excluding small topological subgraphs. In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1411-1434. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.59.
  33. Joseph J. Rotman. An introduction to the theory of groups, volume 148. Springer Science & Business Media, 2012. Google Scholar
  34. V. Vikraman Arvind and Peter J. Cameron. Recognizing the commuting graph of a finite group. arXiv preprint, 2022. URL: https://doi.org/10.48550/arXiv.2206.01059.
  35. Douglas B. West. Introduction to Graph Theory. Prentice Hall, September 2000. Google Scholar
  36. Daniel Wiebking. Graph isomorphism in quasipolynomial time parameterized by treewidth. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. Google Scholar
  37. Samir Zahirović, Ivica Bošnjak, and Rozália Madarász. A study of enhanced power graphs of finite groups. Journal of Algebra and Its Applications, 19(04):2050062, 2020. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact