On the Weisfeiler–Leman Dimension of Permutation Graphs
Abstract.
It is proved that the Weisfeiler–Leman dimension of the class of permutation graphs is at most 18. Previously, it was only known that this dimension is finite (B. Grußien, Proceedings of the 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), 2017, pp. 1–12).
Keywords
MSC codes
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments.
The authors are grateful to the anonymous referees for their remarks that helped to improve the presentation.
References
1.
V. Arvind, J. Köbler, G. Rattan, and O. Verbitsky, Graph isomorphism, color refinement, and compactness, Comput. Complex., 26 (2017), pp. 627–685.
2.
A. Atserias and E. N. Maneva, Sherali–Adams relaxations and indistinguishability in counting logics, SIAM J. Comput., 42 (2013), pp. 112–137.
3.
L. Babai, Group, graphs, algorithms: The graph isomorphism problem, in Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018, Vol. IV, Invited lectures, World Scientific Publishing, 2015, pp. 3319–3336.
4.
L. Babai, Graph isomorphism in quasipolynomial time [extended abstract], in Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, 2016, pp. 684–697.
5.
K. A. Baker, P. C. Fishburn, and F. S. Roberts, Partial orders of dimension 2, Networks, 2 (1971), pp. 11–28.
6.
J.-Y. Cai, M. Fürer, and N. Immerman, An optimal lower bound on the number of variables for graph identification, Combinatorica, 12 (1992), pp. 389–410.
7.
G. Chen and I. Ponomarenko, Coherent Configurations, Central China Normal University Press, Wuhan, China, 2019, https://www.pdmi.ras.ru/∼inp/ccNOTES.pdf (access date: 05/06/2024).
8.
C. J. Colbourn, On testing isomorphism of permutation graphs, Networks, 11 (1981), pp. 13–21.
9.
Conference, Algebraic Graph Theory, Symmetry vs Regularity. The First 50 Years since Weisfeiler–Leman Stabilization, July 1–July 7, 2018, Pilsen, Czech Republic, https://www.iti.zcu.cz/wl2018/index.html (access date: 05/06/2024).
10.
S. Evdokimov and I. Ponomarenko, Separability number and Schurity number of coherent configurations, Electronic J. Combin., 7 (2000), R31.
11.
S. Evdokimov, I. Ponomarenko, and G. Tinhofer, Forestal algebras and algebraic forests (on a new class of weakly compact graphs), Discrete Math., 225 (2000), pp. 149–172.
12.
F. Fuhlbrück, J. Köbler, and O. Verbitsky, Identifiability of graphs with small color classes by the Weisfeiler–Leman algorithm, SIAM J. Discrete Math., 35 (2021), pp. 1792–1853.
13.
T. Gallai, Transitiv orientierbare graphen, Acta Mathematica Academiae Scientiarum Hungarica, 18 (1967), pp. 25–66.
14.
A. L. Gavrilyuk, R. Nedela, and I. Ponomarenko, The Weisfeiler–Leman dimension of distance-hereditary graphs, Graphs Combin., 39 (2023), 84.
15.
M. C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Ann. Discrete Math. 57, Elsevier, Amsterdam, 2004.
16.
M. Grohe, Fixed-point definability and polynomial time on graphs with excluded minors, in Proceedings of the 25th Annual IEEE Symposium on Logic in Computer Science (LICS), 2010, pp. 179–188.
17.
M. Grohe, Descriptive Complexity, Canonisation and Definable Graph Structure Theory, Cambridge University Press, Cambridge, UK, 2017.
18.
M. Grohe and S. Kiefer, A linear upper bound on the Weisfeiler–Leman dimension of graphs of bounded genus, in Proceedings of the 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019), Dagstühl Publishing, 2019, pp. 117:1–117:15, https://doi.org/10.4230/LIPIcs.ICALP.2019.117.
19.
M. Grohe and D. Neuen, Canonisation and definability for graphs of bounded rank width, in Proceedings of the 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), 2019, pp. 1–13, https://doi.org/10.1109/LICS.2019.8785682.
20.
M. Grohe and M. Otto, Pebble games and linear equations, in CSL, LIPIcs 16, 2012, pp. 289–304.
21.
B. Grußien, Capturing polynomial time using modular decomposition, in Proceedings of the 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), 2017, pp. 1–12, https://doi.org/10.1109/LICS.2017.8005086.
22.
N. Immerman and E. Lander, Describing graphs: A first-order approach to graph canonization, in Complexity Theory Retrospective, A. Selman, ed., Springer, New York, 1990, pp. 59–81.
23.
S. Kiefer, I. Ponomarenko, and P. Schweitzer, The Weisfeiler–Leman dimension of planar graphs is at most 3, J. ACM, 66 (2019), 44, https://doi.org/10.1145/3333003.
24.
S. Kiefer, P. Schweitzer, and E. Selman, Graphs identified by logics with counting, ACM Trans. Comput. Log., 23 (2022), pp. 1–31.
25.
A. Pnueli, A. Lempel, and S. Even, Transitive orientation of graphs and identification of permutation graphs, Canadian J. Math., 23 (1971), pp. 160–175.
26.
I. Ponomarenko and G. Ryabov, The Weisfeiler–Leman dimension of chordal bipartite graphs without bipartite claw, Graphs Combin., 37 (2021), pp. 1089–1102.
27.
J. Spinrad, On comparability and permutation graphs, SIAM J. Comput., 14 (1985), pp. 658–670.
28.
B. Weisfeiler and A. Leman, The reduction of a graph to canonical form and the algebra which appears therein, Nauchno-Technicheskaya Informatsia, 2 (1968), pp. 12–16.
Information & Authors
Information
Published In
SIAM Journal on Discrete Mathematics
Pages: 1915 - 1929
DOI: 10.1137/23M1575019
ISSN (online): 1095-7146
Copyright
© 2024 the authors.
History
Submitted: 24 May 2023
Accepted: 25 March 2024
Published online: 21 June 2024
Keywords
MSC codes
Authors
Funding Information
Funding: The research of the first author is supported by the National Natural Science Foundation of China, grant 12361003. The research of the second author is supported by JSPS KAKENHI grant 22K03403.
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
There are no citations for this item
View Options
Get Access
- Access via your Institution
- Questions about how to access this content? Contact SIAM at [email protected].