Coloring graphs with locally few colors
Discrete Mathematics
Languages that capture complexity classes
SIAM Journal on Computing
Chromatically optimal rigid graphs
Journal of Combinatorial Theory Series B
Complexity of graph partition problems
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
On the Structure of Polynomial Time Reducibility
Journal of the ACM (JACM)
Duality theorems for finite structures (characterising gaps and good characterisations)
Journal of Combinatorial Theory Series B
The complexity of relational query languages (Extended Abstract)
STOC '82 Proceedings of the fourteenth annual ACM symposium on Theory of computing
On sparse graphs with given colorings and homomorphisms
Journal of Combinatorial Theory Series B
On Digraph Coloring Problems and Treewidth Duality
LICS '05 Proceedings of the 20th Annual IEEE Symposium on Logic in Computer Science
Existential Positive Types and Preservation under Homomorphisisms
LICS '05 Proceedings of the 20th Annual IEEE Symposium on Logic in Computer Science
On preservation under homomorphisms and unions of conjunctive queries
Journal of the ACM (JACM)
Linear time low tree-width partitions and algorithmic consequences
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
A Probabilistic Approach to the Dichotomy Problem
SIAM Journal on Computing
European Journal of Combinatorics
Forbidden lifts (NP and CSP for combinatorialists)
European Journal of Combinatorics
Grad and classes with bounded expansion III. Restricted graph homomorphism dualities
European Journal of Combinatorics
On the CSP dichotomy conjecture
CSR'11 Proceedings of the 6th international conference on Computer science: theory and applications
SIAM Journal on Discrete Mathematics
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We show that every NP problem is polynomially equivalent to a simple combinatorial problem: the membership problem for a special class of digraphs. These classes are defined by means of shadows (projections) and by finitely many forbidden colored (lifted) subgraphs. Our characterization is motivated by the analysis of syntactical subclasses with the full computational power of NP, which were first studied by Feder and Vardi. Our approach applies to many combinatorial problems and it induces the characterization of coloring problems (CSP) defined by means of shadows. This turns out to be related to homomorphism dualities. We prove that a class of digraphs (relational structures) defined by finitely many forbidden colored subgraphs (i.e. lifted substructures) is a CSP class if and only if all the the forbidden structures are homomorphically equivalent to trees. We show a surprising richness of coloring problems when restricted to most frequent graph classes. Using results of Nešetřil and Ossona de Mendez for bounded expansion classes (which include bounded degree and proper minor closed classes) we prove that the restriction of every class defined as the shadow of finitely many colored subgraphs equals to the restriction of a coloring (CSP) class.