Homomorphism-Distinguishing Closedness for Graphs of Bounded Tree-Width

Author Daniel Neuen

Thumbnail PDF


  • Filesize: 0.69 MB
  • 12 pages

Document Identifiers

Author Details

Daniel Neuen
  • University of Bremen, Germany


I want to thank Tim Seppelt for pointing me to Lemma 4 in [Seppelt, MFCS 2023].

Cite AsGet BibTex

Daniel Neuen. Homomorphism-Distinguishing Closedness for Graphs of Bounded Tree-Width. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 53:1-53:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


Two graphs are homomorphism indistinguishable over a graph class 𝐅, denoted by G ≡_𝐅 H, if hom(F,G) = hom(F,H) for all F ∈ 𝐅 where hom(F,G) denotes the number of homomorphisms from F to G. A classical result of Lovász shows that isomorphism between graphs is equivalent to homomorphism indistinguishability over the class of all graphs. More recently, there has been a series of works giving natural algebraic and/or logical characterizations for homomorphism indistinguishability over certain restricted graph classes. A class of graphs 𝐅 is homomorphism-distinguishing closed if, for every F ∉ 𝐅, there are graphs G and H such that G ≡_𝐅 H and hom(F,G) ≠ hom(F,H). Roberson conjectured that every class closed under taking minors and disjoint unions is homomorphism-distinguishing closed which implies that every such class defines a distinct equivalence relation between graphs. In this work, we confirm this conjecture for the classes 𝒯_k, k ≥ 1, containing all graphs of tree-width at most k. As an application of this result, we also characterize which subgraph counts are detected by the k-dimensional Weisfeiler-Leman algorithm. This answers an open question from [Arvind et al., J. Comput. Syst. Sci., 2020].

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Mathematics of computing → Combinatorial algorithms
  • Theory of computation → Graph algorithms analysis
  • homomorphism indistinguishability
  • tree-width
  • Weisfeiler-Leman algorithm
  • subgraph counts


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


  1. Vikraman Arvind, Frank Fuhlbrück, Johannes Köbler, and Oleg Verbitsky. On Weisfeiler-Leman invariance: Subgraph counts and related graph properties. J. Comput. Syst. Sci., 113:42-59, 2020. URL: https://doi.org/10.1016/j.jcss.2020.04.003.
  2. Jin-yi Cai, Martin Fürer, and Neil Immerman. An optimal lower bound on the number of variables for graph identification. Comb., 12(4):389-410, 1992. URL: https://doi.org/10.1007/BF01305232.
  3. Zhengdao Chen, Lei Chen, Soledad Villar, and Joan Bruna. Can graph neural networks count substructures? In Hugo Larochelle, Marc'Aurelio Ranzato, Raia Hadsell, Maria-Florina Balcan, and Hsuan-Tien Lin, editors, Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020, NeurIPS 2020, December 6-12, 2020, virtual, 2020. URL: https://proceedings.neurips.cc/paper/2020/hash/75877cb75154206c4e65e76b88a12712-Abstract.html.
  4. Radu Curticapean, Holger Dell, and Dániel Marx. Homomorphisms are a good basis for counting small subgraphs. In Hamed Hatami, Pierre McKenzie, and Valerie King, editors, Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017, pages 210-223. ACM, 2017. URL: https://doi.org/10.1145/3055399.3055502.
  5. Anuj Dawar and David Richerby. The power of counting logics on restricted classes of finite structures. In Jacques Duparc and Thomas A. Henzinger, editors, Computer Science Logic, 21st International Workshop, CSL 2007, 16th Annual Conference of the EACSL, Lausanne, Switzerland, September 11-15, 2007, Proceedings, volume 4646 of Lecture Notes in Computer Science, pages 84-98. Springer, 2007. URL: https://doi.org/10.1007/978-3-540-74915-8_10.
  6. Holger Dell, Martin Grohe, and Gaurav Rattan. Lovász meets Weisfeiler and Leman. In Ioannis Chatzigiannakis, Christos Kaklamanis, Dániel Marx, and Donald Sannella, editors, 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, July 9-13, 2018, Prague, Czech Republic, volume 107 of LIPIcs, pages 40:1-40:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.ICALP.2018.40.
  7. Banu Dost, Tomer Shlomi, Nitin Gupta, Eytan Ruppin, Vineet Bafna, and Roded Sharan. QNet: a tool for querying protein interaction networks. J. Comput. Biol., 15(7):913-925, 2008. URL: https://doi.org/10.1089/cmb.2007.0172.
  8. Zdeněk Dvořák. On recognizing graphs by numbers of homomorphisms. J. Graph Theory, 64(4):330-342, 2010. URL: https://doi.org/10.1002/jgt.20461.
  9. Eva Fluck, Tim Seppelt, and Gian Luca Spitzer. Going deep and going wide: Counting logic and homomorphism indistinguishability over graphs of bounded treedepth and treewidth. CoRR, abs/2308.06044, 2023. URL: https://doi.org/10.48550/arXiv.2308.06044.
  10. Martin Fürer. On the combinatorial power of the Weisfeiler-Lehman algorithm. In Dimitris Fotakis, Aris Pagourtzis, and Vangelis Th. Paschos, editors, Algorithms and Complexity - 10th International Conference, CIAC 2017, Athens, Greece, May 24-26, 2017, Proceedings, volume 10236 of Lecture Notes in Computer Science, pages 260-271, 2017. URL: https://doi.org/10.1007/978-3-319-57586-5_22.
  11. Andreas Göbel, Leslie Ann Goldberg, and Marc Roth. The Weisfeiler-Leman dimension of existential conjunctive queries. CoRR, abs/2310.19006, 2023. URL: https://doi.org/10.48550/arXiv.2310.19006.
  12. Martin Grohe. Counting bounded tree depth homomorphisms. In Holger Hermanns, Lijun Zhang, Naoki Kobayashi, and Dale Miller, editors, LICS '20: 35th Annual ACM/IEEE Symposium on Logic in Computer Science, Saarbrücken, Germany, July 8-11, 2020, pages 507-520. ACM, 2020. URL: https://doi.org/10.1145/3373718.3394739.
  13. Martin Grohe, Gaurav Rattan, and Tim Seppelt. Homomorphism tensors and linear equations. In Mikolaj Bojanczyk, Emanuela Merelli, and David P. Woodruff, editors, 49th International Colloquium on Automata, Languages, and Programming, ICALP 2022, July 4-8, 2022, Paris, France, volume 229 of LIPIcs, pages 70:1-70:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ICALP.2022.70.
  14. Falk Hüffner, Sebastian Wernicke, and Thomas Zichner. Algorithm engineering for color-coding with applications to signaling pathway detection. Algorithmica, 52(2):114-132, 2008. URL: https://doi.org/10.1007/s00453-007-9008-7.
  15. Matthias Lanzinger and Pablo Barceló. On the power of the Weisfeiler-Leman test for graph motif parameters. CoRR, abs/2309.17053, 2023. URL: https://doi.org/10.48550/arXiv.2309.17053.
  16. László Lovász. Operations with structures. Acta Math. Acad. Sci. Hungar., 18:321-328, 1967. URL: https://doi.org/10.1007/BF02280291.
  17. Laura Mancinska and David E. Roberson. Quantum isomorphism is equivalent to equality of homomorphism counts from planar graphs. In Sandy Irani, editor, 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020, pages 661-672. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00067.
  18. Ron Milo, Shai S. Shen-Orr, Shalev Itzkovitz, Nadav Kashtan, Dmitri B. Chklovskii, and Uri Alon. Network motifs: Simple building blocks of complex networks. Science, 298(5594):824-827, 2002. URL: https://doi.org/10.1126/science.298.5594.824.
  19. Christopher Morris, Yaron Lipman, Haggai Maron, Bastian Rieck, Nils M. Kriege, Martin Grohe, Matthias Fey, and Karsten M. Borgwardt. Weisfeiler and Leman go machine learning: The story so far. CoRR, abs/2112.09992, 2021. URL: https://arxiv.org/abs/2112.09992.
  20. Christopher Morris, Martin Ritzert, Matthias Fey, William L. Hamilton, Jan Eric Lenssen, Gaurav Rattan, and Martin Grohe. Weisfeiler and Leman go neural: Higher-order graph neural networks. In The Thirty-Third AAAI Conference on Artificial Intelligence, AAAI 2019, The Thirty-First Innovative Applications of Artificial Intelligence Conference, IAAI 2019, The Ninth AAAI Symposium on Educational Advances in Artificial Intelligence, EAAI 2019, Honolulu, Hawaii, USA, January 27 - February 1, 2019, pages 4602-4609. AAAI Press, 2019. URL: https://doi.org/10.1609/aaai.v33i01.33014602.
  21. David E. Roberson. Oddomorphisms and homomorphism indistinguishability over graphs of bounded degree. CoRR, abs/2206.10321, 2022. URL: https://doi.org/10.48550/arXiv.2206.10321.
  22. David E. Roberson and Tim Seppelt. Lasserre hierarchy for graph isomorphism and homomorphism indistinguishability. In Kousha Etessami, Uriel Feige, and Gabriele Puppis, editors, 50th International Colloquium on Automata, Languages, and Programming, ICALP 2023, July 10-14, 2023, Paderborn, Germany, volume 261 of LIPIcs, pages 101:1-101:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.ICALP.2023.101.
  23. Jacob Scott, Trey Ideker, Richard M. Karp, and Roded Sharan. Efficient algorithms for detecting signaling pathways in protein interaction networks. J. Comput. Biol., 13(2):133-144, 2006. URL: https://doi.org/10.1089/cmb.2006.13.133.
  24. Tim Seppelt. Logical equivalences, homomorphism indistinguishability, and forbidden minors. In Jérôme Leroux, Sylvain Lombardy, and David Peleg, editors, 48th International Symposium on Mathematical Foundations of Computer Science, MFCS 2023, August 28 to September 1, 2023, Bordeaux, France, volume 272 of LIPIcs, pages 82:1-82:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.MFCS.2023.82.
  25. Paul D. Seymour and Robin Thomas. Graph searching and a min-max theorem for tree-width. J. Comb. Theory, Ser. B, 58(1):22-33, 1993. URL: https://doi.org/10.1006/jctb.1993.1027.
  26. Nino Shervashidze, Pascal Schweitzer, Erik Jan van Leeuwen, Kurt Mehlhorn, and Karsten M. Borgwardt. Weisfeiler-Lehman graph kernels. J. Mach. Learn. Res., 12:2539-2561, 2011. URL: https://doi.org/10.5555/1953048.2078187.
  27. Tomer Shlomi, Daniel Segal, Eytan Ruppin, and Roded Sharan. Qpath: a method for querying pathways in a protein-protein interaction network. BMC Bioinformatics, 7:199, 2006. URL: https://doi.org/10.1186/1471-2105-7-199.
  28. Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka. How powerful are graph neural networks? In 7th International Conference on Learning Representations, ICLR 2019, New Orleans, LA, USA, May 6-9, 2019. OpenReview.net, 2019. URL: https://openreview.net/forum?id=ryGs6iA5Km.