On the complexity of H-coloring
Journal of Combinatorial Theory Series B
Vertex partitioning problems: characterization, complexity and algorithms on partial K-trees
Vertex partitioning problems: characterization, complexity and algorithms on partial K-trees
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Algorithms for Vertex Partitioning Problems on Partial k-Trees
SIAM Journal on Discrete Mathematics
Partitioning graphs into generalized dominating sets
Nordic Journal of Computing
Complexity of domination-type problems in graphs
Nordic Journal of Computing
Generalized domination in chordal graphs
Nordic Journal of Computing
The complexity of satisfiability problems
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
Classifying the Complexity of Constraints Using Finite Algebras
SIAM Journal on Computing
A complete complexity classification of the role assignment problem
Theoretical Computer Science - Graph colorings
Bi-arc graphs and the complexity of list homomorphisms
Journal of Graph Theory
Locally injective graph homomorphism: lists guarantee dichotomy
WG'06 Proceedings of the 32nd international conference on Graph-Theoretic Concepts in Computer Science
Generalized domination in degenerate graphs: a complete dichotomy of computational complexity
TAMC'08 Proceedings of the 5th international conference on Theory and applications of models of computation
Parameterized complexity of generalized domination problems
Discrete Applied Mathematics
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The so called (σ, ρ)-domination, introduced by J.A. Telle, is a concept which provides a unifying generalization for many variants of domination in graphs. (A set S of vertices of a graph G is called (σ, ρ)- dominating if for every vertex v ∈ S, |S ∩ N(v)|∈ σ, and for every v∉S, |S ∩ N(v)| ∈ ρ, where σ and ρ are sets of nonnegative integers and N(v) denotes the open neighborhood of the vertex v in G). It was known that for any two nonempty finite sets σ and ρ (such that 0 ∉ ρ), the decision problem whether an input graph contains a (σ, ρ)-dominating set is NP-complete, but that when restricted to chordal graphs, some polynomial time solvable instances occur. We show that for chordal graphs, the problem performs a complete dichotomy: it is polynomial time solvable if σ, ρ are such that every chordal graph contains at most one (σ, ρ)- dominating set, and NP-complete otherwise. The proof involves certain flavor of existentionality - we are not able to characterize such pairs (σ, ρ) by a structural description, but at least we can provide a recursive algorithm for their recognition. If ρ contains the 0 element, every graph contains a (σ, ρ)-dominating set (the empty one), and so the nontrivial question here is to ask for a maximum such set. We show that MAX- (σ, ρ)-domination problem is NP-complete for chordal graphs whenever ρ contains, besides 0, at least one more integer.