Tight Bounds on Adjacency Labels for Monotone Graph Classes

Authors Édouard Bonnet , Julien Duron , John Sylvester , Viktor Zamaraev , Maksim Zhukovskii



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Author Details

Édouard Bonnet
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
Julien Duron
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
John Sylvester
  • Department of Computer Science, University of Liverpool, UK
Viktor Zamaraev
  • Department of Computer Science, University of Liverpool, UK
Maksim Zhukovskii
  • Department of Computer Science, University of Sheffield, UK

Acknowledgements

We are grateful to Nathan Harms for valuable feedback on the early version of this paper.

Cite AsGet BibTex

Édouard Bonnet, Julien Duron, John Sylvester, Viktor Zamaraev, and Maksim Zhukovskii. Tight Bounds on Adjacency Labels for Monotone Graph Classes. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 31:1-31:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.31

Abstract

A class of graphs admits an adjacency labeling scheme of size b(n), if the vertices in each of its n-vertex graphs can be assigned binary strings (called labels) of length b(n) so that the adjacency of two vertices can be determined solely from their labels. We give bounds on the size of adjacency labels for every family of monotone (i.e., subgraph-closed) classes with a "well-behaved" growth function between 2^Ω(n log n) and 2^O(n^{2-δ}) for any δ > 0. Specifically, we show that for any function f: ℕ → ℝ satisfying log n ⩽ f(n) ⩽ n^{1-δ} for any fixed δ > 0, and some sub-multiplicativity condition, there are monotone graph classes with growth 2^O(nf(n)) that do not admit adjacency labels of size at most f(n) log n. On the other hand, any such class does admit adjacency labels of size O(f(n)log n). Surprisingly this bound is a Θ(log n) factor away from the information-theoretic bound of Ω(f(n)). Our bounds are tight upto constant factors, and the special case when f = log implies that the recently-refuted Implicit Graph Conjecture [Hatami and Hatami, FOCS 2022] also fails within monotone classes. We further show that the Implicit Graph Conjecture holds for all monotone small classes. In other words, any monotone class with growth rate at most n! cⁿ for some constant c > 0, admits adjacency labels of information-theoretic order optimal size. In fact, we show a more general result that is of independent interest: any monotone small class of graphs has bounded degeneracy. We conjecture that the Implicit Graph Conjecture holds for all hereditary small classes.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics
  • Mathematics of computing → Graph theory
Keywords
  • Adjacency labeling
  • degeneracy
  • monotone classes
  • small classes
  • factorial classes
  • implicit graph conjecture

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References

  1. Vladimir E. Alekseev. Range of values of entropy of hereditary classes of graphs. Diskretnaya Matematika, 4(2):148-157, 1992. Google Scholar
  2. Vladimir E. Alekseev. On lower layers of a lattice of hereditary classes of graphs. Diskretnyi Analiz i Issledovanie Operatsii, 4(1):3-12, 1997. Google Scholar
  3. Noga Alon. Asymptotically optimal induced universal graphs. Geom. Funct. Anal., 27(1):1-32, 2017. URL: https://doi.org/10.1007/s00039-017-0396-9.
  4. Noga Alon. Implicit representation of sparse hereditary families. Discret. Comput. Geom., to appear, 2023. URL: https://doi.org/10.1007/s00454-023-00521-0.
  5. Noga Alon and Joel H. Spencer. The Probabilistic Method, Third Edition. Wiley-Interscience series in discrete mathematics and optimization. Wiley, 2008. Google Scholar
  6. Stephen Alstrup, Haim Kaplan, Mikkel Thorup, and Uri Zwick. Adjacency labeling schemes and induced-universal graphs. In Proceedings of the forty-seventh annual ACM symposium on Theory of Computing, pages 625-634, 2015. Google Scholar
  7. Avah Banerjee. An adjacency labeling scheme based on a decomposition of trees into caterpillars. In Combinatorial Algorithms - 33rd International Workshop, IWOCA 2022, volume 13270 of Lecture Notes in Computer Science, pages 114-127. Springer, 2022. Google Scholar
  8. Robin L. Blankenship. Book embeddings of graphs. Louisiana State University and Agricultural & Mechanical College, 2003. Google Scholar
  9. Béla Bollobás and Andrew Thomason. Projections of bodies and hereditary properties of hypergraphs. Bulletin of the London Mathematical Society, 27(5):417-424, 1995. Google Scholar
  10. Marthe Bonamy, Louis Esperet, Carla Groenland, and Alex Scott. Optimal labelling schemes for adjacency, comparability, and reachability. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 1109-1117, 2021. Google Scholar
  11. Édouard Bonnet, Julien Duron, John Sylvester, Viktor Zamaraev, and Maksim Zhukovskii. Tight bounds on adjacency labels for monotone graph classes. arXiv:2310.20522, 2023. Google Scholar
  12. Édouard Bonnet, Julien Duron, John Sylvester, Viktor Zamaraev, and Maksim Zhukovskii. Small but unwieldy: A lower bound on adjacency labels for small classes. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2024), 2024. Google Scholar
  13. Édouard Bonnet, Colin Geniet, Eun Jung Kim, Stéphan Thomassé, and Rémi Watrigant. Twin-width II: small classes. Combinatorial Theory, 2 (2), 2022. Google Scholar
  14. Maurice Chandoo. Logical labeling schemes. Discrete Mathematics, 346(10):113565, 2023. Google Scholar
  15. Bruno Courcelle and Rémi Vanicat. Query efficient implementation of graphs of bounded clique-width. Discrete Applied Mathematics, 131(1):129-150, 2003. Google Scholar
  16. Louis Esperet, Nathaniel Harms, and Andrey Kupavskii. Sketching distances in monotone graph classes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. Google Scholar
  17. Louis Esperet, Nathaniel Harms, and Viktor Zamaraev. Optimal adjacency labels for subgraphs of cartesian products. In 50th International Colloquium on Automata, Languages, and Programming, ICALP 2023, volume 261 of LIPIcs, pages 57:1-57:11. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.ICALP.2023.57.
  18. Pierre Fraigniaud and Amos Korman. On randomized representations of graphs using short labels. In SPAA 2009: Proceedings of the 21st Annual ACM Symposium on Parallelism in Algorithms and Architectures, pages 131-137. ACM, 2009. URL: https://doi.org/10.1145/1583991.1584031.
  19. Alan Frieze and Michał Karoński. Random Graphs and Networks: A First Course. Cambridge University Press, 2023. URL: https://doi.org/10.1017/9781009260268.
  20. Cyril Gavoille and Arnaud Labourel. Shorter implicit representation for planar graphs and bounded treewidth graphs. In 15th Annual European Symposium on Algorithms, ESA 2007, volume 4698 of Lecture Notes in Computer Science, pages 582-593. Springer, 2007. Google Scholar
  21. Nathaniel Harms. Universal communication, universal graphs, and graph labeling. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. Google Scholar
  22. Nathaniel Harms, Sebastian Wild, and Viktor Zamaraev. Randomized communication and implicit graph representations. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing (STOC 2022), pages 1220-1233, 2022. Google Scholar
  23. Hamed Hatami and Pooya Hatami. The implicit graph conjecture is false. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS 2022), pages 1134-1137. IEEE, 2022. Google Scholar
  24. Sampath Kannan, Moni Naor, and Steven Rudich. Implicit representation of graphs. In Proceedings of the twentieth annual ACM symposium on Theory of computing (STOC 1988), pages 334-343, 1988. Google Scholar
  25. Sampath Kannan, Moni Naor, and Steven Rudich. Implicit representation of graphs. SIAM Journal on Discrete Mathematics, 5(4):596-603, 1992. Google Scholar
  26. Ilia Krasikov, Arieh Lev, and Bhalchandra D Thatte. Upper bounds on the automorphism group of a graph0. Discrete Mathematics, 256(1-2):489-493, 2002. Google Scholar
  27. John W Moon. On minimal n-universal graphs. Glasgow Mathematical Journal, 7(1):32-33, 1965. Google Scholar
  28. John H. Muller. Local Structure in Graph Classes. PhD thesis, Georgia Institute of Technology, 1988. Google Scholar
  29. Serguei Norine, Paul Seymour, Robin Thomas, and Paul Wollan. Proper minor-closed families are small. Journal of Combinatorial Theory, Series B, 96(5):754-757, 2006. Google Scholar
  30. Edward R Scheinerman. Local representations using very short labels. Discrete mathematics, 203(1-3):287-290, 1999. Google Scholar
  31. Edward R Scheinerman and Jennifer Zito. On the size of hereditary classes of graphs. Journal of Combinatorial Theory, Series B, 61(1):16-39, 1994. Google Scholar
  32. Jeremy P Spinrad. Efficient Graph Representations, volume 19. Fields Institute monographs. American Mathematical Soc., 2003. Google Scholar