Abstract
We study the class of all finite directed graphs (digraphs) up to primitive positive constructibility. The resulting order has a unique maximal element, namely the digraph P1 with one vertex and no edges. The digraph P1 has a unique maximal lower bound, namely the digraph P2 with two vertices and one directed edge. Our main result is a complete description of the maximal lower bounds of P2; we call these digraphs submaximal. We show that every digraph that is not equivalent to P1 and P2 is below one of the submaximal digraphs.
This is a preview of subscription content, access via your institution.
References
J. Nešetřil and C. Tardif: Duality theorems for finite structures, Journal of Combinatorial Theory, Series B 80 (2000), 80–97.
P. Hell and J. Nešetřil: On the complexity of H-coloring, Journal of Combinatorial Theory, Series B 48 (1990), 92–110.
T. Feder and M. Y. Vardi: The computational structure of monotone monadic SNP and constraint satisfaction: a study through Datalog and group theory, SIAM Journal on Computing 28 (1999), 57–104.
J. Bulin, D. Delic, M. Jackson and T. Niven: A finer reduction of constraint problems to digraphs, Log. Methods Comput. Sci. 11 (2015).
A. A. Bulatov: A dichotomy theorem for nonuniform CSPs, in: 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15–17, 319–330, 2017.
D. Zhuk: A proof of the CSP dichotomy conjecture, J. ACM 67 (2020), 1–78.
V. Dalmau: Linear datalog and bounded path duality of relational structures, Logical Methods in Computer Science 1 (2005).
L. Egri, B. Larose and P. Tesson: Symmetric datalog and constraint satisfaction problems in logspace, in: Proceedings of the Symposium on Logic in Computer Science (LICS), 193–202, 2007.
A. Kazda: n-permutability and linear Datalog implies symmetric Datalog, Log. Methods Comput. Sci. 14 (2018).
L. Barto, J. Opršal and M. Pinsker: The wonderland of reflections, Israel Journal of Mathematics 223 (2018), 363–398.
M. Bodirsky, F. Starke and A. Vucaj: Smooth digraphs modulo primitive positive constructability and cyclic loop conditions. International Journal on Algebra and Computation 31 (2021), 939–967.
M. Bodirsky and A. Vucaj: Two-element structures modulo primitive positive constructability, Algebra Universalis, 81, 2020.
P. Hell and J. Nešetřil: Graphs and Homomorphisms, Oxford University Press, Oxford, 2004.
M. Bodirsky: Cores of countably categorical structures, Logical Methods in Computer Science (LMCS) 3 (2007), 1–16.
C. Carvalho, L. Egri, M. Jackson and T. Niven: On Maltsev digraphs, Electr. J. Comb. 22 (2015), P1.47.
B. L. Bauslaugh: Cores and compactness of infinite directed graphs, Journal of Combinatorial Theory, Series B 68 (1996), 255–276.
M. Bodirsky, A. Mottet, M. Olšák, J. Opršal, M. Pinsker and R. Willard: ω-categorical structures avoiding height 1 identities, Transactions of the American Mathematical Society, accepted.
M. Jackson, T. Kowalski and T. Niven: Complexity and polymorphisms for digraph constraint problems under some basic constructions, Int. J. Algebra Comput. 26 (2016), 1395–1433.
Acknowledgement
The authors would like to thank the anonymous referees for thoroughly reading our article as well as giving helpful comments that improved the final article.
Author information
Authors and Affiliations
Corresponding author
Additional information
The second author is supported by DFG Graduiertenkolleg 1763 (QuantLA).
Rights and permissions
About this article
Cite this article
Bodirsky, M., Starke, F. Maximal Digraphs with Respect to Primitive Positive Constructability. Combinatorica 42 (Suppl 1), 997–1010 (2022). https://doi.org/10.1007/s00493-022-4918-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00493-022-4918-1
Mathematics Subject Classification (2010)
- 05C15
- 05C20