Dejan Delić

On the reduction of the CSP dichotomy conjecture to digraphs

It is well known that the constraint satisfaction problem over general relational structures can be reduced in polynomial time to digraphs. We present a simple variant of such a reduction and use it to show that the algebraic dichotomy conjecture is equivalent to its restriction to digraphs and that the polynomial reduction can be made in logspace. We also show that our reduction preserves the bounded width property, i.e., solvability by local consistency methods. We discuss further algorithmic properties that are preserved and related open problems. The first author was supported by the grant projects GACR 201/09/H012, GA UK 67410, SVV-2013-267317; the second author gratefully acknowledges support by the Natural Sciences and Engineering Research Council of Canada in the form of a Discovery Grant; the third and fourth were supported by ARC Discovery Project DP1094578; the first and fourth authors were also supported by the Fields Institute.

The Monoid of the Random Graph

R. We show that End(R) is not regular and is not generated by its idempotents. The Rees order on the idempotents of End(R) has 2N0 many minimal elements. We also prove that the order type of Q is embeddable in the Rees order of End(R).

A finer reduction of constraint problems to digraphs

It is well known that the constraint satisfaction problem over a general relational structure A is polynomial time equivalent to the constraint problem over some associated digraph. We present a variant of this construction and show that the corresponding constraint satisfaction problem is logspace equivalent to that over A. Moreover, we show that almost all of the commonly encountered polymorphism properties are held equivalently on the A and the constructed digraph. As a consequence, the Algebraic CSP dichotomy conjecture as well as the conjectures characterizing CSPs solvable in logspace and in nondeterministic logspace are equivalent to their restriction to digraphs.

On a Problem of Cameron’s on Inexhaustible Graphs

A graph G is inexhaustible if whenever a vertex of G is deleted the remaining graph is isomorphic to G. We address a question of Cameron [6], who asked which countable graphs are inexhaustible. In particular, we prove that there are continuum many countable inexhaustible graphs with properties in common with the infinite random graph, including adjacency properties and universality. Locally finite inexhaustible graphs and forests are investigated, as is a semigroup structure on the class of inexhaustible graphs. We extend a result of [7] on homogeneous inexhaustible graphs to pseudo-homogeneous inexhaustible graphs.

All countable monoids embed into the monoid of the infinite random graph

We prove that the full transformation monoid on a countably infinite set is isomorphic to a submonoid of End(R), the endomorphism monoid of the infinite random graph R. Consequently, End(R) embeds each countable monoid, satisfies no nontrivial monoid identity, and has an undecidable universal theory.

Generalized Pigeonhole Properties of Graphs and Oriented Graphs

A relational structure A satisfies the P(n, k) property if whenever the vertex set of A is partitioned into n nonempty parts, the substructure induced by the union of somek of the parts is isomorphic to A. The P(2, 1) property is just the pigeonhole property, (P), introduced by Cameron, and studied by Bonato, Deli? and Cameron. We classify the countable graphs, tournaments, and oriented graphs with the P(3, 2) property.

A finitely axiomatizable undecidable equational theory with recursively solvable word problems

In this paper we construct a finitely based variety, whose equational theory is undecidable, yet whose word problems are recursively solvable, which solves a problem stated by G. McNulty (1992). The construction produces a discriminator variety with the aforementioned properties starting from a class of structures in some multisorted language (which may include relations), axiomatized by a finite set of universal sentences in the given multisorted signature. This result also presents a common generalization of the earlier results obtained by B. Wells (1982) and A. Mekler, E. Nelson, and S. Shelah (1993).

A note on orientations of the infinite random graph

We answer a question of Camerons by giving examples of 2N 0 many non-isomorphic acyclic orientations of the infinite random graph with a topological ordering that do not have the pigeonhole property. Our examples also embed each countable linear ordering.

SOLUTION TO A PROBLEM OF KUBLANOVSKY AND SAPIR

We construct a finitely based congruence-distributive variety of algebras in a finite language whose set of subalgebras of finite simple algebras is non-recursive.

The Model Companion of Width-Two Orders

The class of width-two orders has a model companion. The model companion is complete, decidable, non-finitely axiomatizable, and has continuum many countable models. Generalizations of some results in (Pouzet, M., 1978, J. Combin. Theory Ser. B 25) are presented in the width-n case, for n2.