Vertex partitioning problems: characterization, complexity and algorithms on partial K-trees
Vertex partitioning problems: characterization, complexity and algorithms on partial K-trees
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
Computational complexity of generalized domination: a complete dichotomy for chordal graphs
WG'07 Proceedings of the 33rd international conference on Graph-theoretic concepts in computer science
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 is 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 some graph classes, polynomial time solvable instances occur. We show that for every k, the problem performs a complete dichotomy when restricted to k-degenerate graphs, and we fully characterize the polynomial and NPcomplete instances. It is further shown that the problem is polynomial time solvable if σ, ρ are such that every k-degenerate graph contains at most one (σ, ρ)-dominating set, and NP-complete otherwise. This relates to the concept of ambivalent graphs previously introduced for chordal graphs.