Maximal Digraphs With Respect to Primitive Positive Constructibility

We study the class of all finite directed graphs (digraphs) up to primitive positive constructability. The resulting order has a unique greatest element, namely the digraph P1 with one vertex and no edges. The digraph P1 has a unique greatest lower bound, namely the digraph P2 with two vertices and one directed edge. Our main result is a complete description of the greatest 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.

[1]  Jaroslav Nesetril,et al.  On the complexity of H-coloring , 1990, J. Comb. Theory, Ser. B.

[2]  Manuel Bodirsky,et al.  Complexity of Infinite-Domain Constraint Satisfaction , 2021 .

[3]  Todd Niven,et al.  Complexity and polymorphisms for digraph constraint problems under some basic constructions , 2016, Int. J. Algebra Comput..

[4]  Dmitriy Zhuk,et al.  A Proof of CSP Dichotomy Conjecture , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[5]  Jaroslav Nesetril,et al.  Graphs and homomorphisms , 2004, Oxford lecture series in mathematics and its applications.

[6]  Andrei A. Bulatov,et al.  A Dichotomy Theorem for Nonuniform CSPs , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[7]  Tomás Feder,et al.  The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory , 1999, SIAM J. Comput..

[8]  Víctor Dalmau,et al.  Linear datalog and bounded path duality of relational structures , 2005, Log. Methods Comput. Sci..

[9]  Alexandr Kazda,et al.  $n$-permutability and linear Datalog implies symmetric Datalog , 2015, Log. Methods Comput. Sci..

[10]  Libor Barto,et al.  The wonderland of reflections , 2015, Israel Journal of Mathematics.

[12]  Todd Niven,et al.  On Maltsev Digraphs , 2011, Electron. J. Comb..

[13]  M. Bodirsky,et al.  Two-element structures modulo primitive positive constructability , 2019, Algebra universalis.

[14]  Pascal Tesson,et al.  Symmetric Datalog and Constraint Satisfaction Problems in Logspace , 2007, 22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007).

[15]  Todd Niven,et al.  A finer reduction of constraint problems to digraphs , 2014, Log. Methods Comput. Sci..