Abstract
Given two sets σ, ρ of nonnegative integers, a set S of vertices of a graph G is (σ,ρ)-dominating if |S ∩ N(v)| ∈ σ for every vertex v ∈ S, and |S ∩ N(v)| ∈ ρ for every v ∉ S. This concept, introduced by Telle in 1990’s, generalizes and unifies several variants of graph domination studied separately before. We study the parameterized complexity of (σ,ρ)-domination in this general setting. Among other results we show that existence of a (σ,ρ)-dominating set of size k (and at most k) are W[1]-complete problems (when parameterized by k) for any pair of finite sets σ and ρ. We further present results on dual parametrization by n − k, and results on certain infinite sets (in particular for σ, ρ being the sets of even and odd integers).
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Golovach, P.A., Kratochvíl, J., Suchý, O. (2010). Parameterized Complexity of Generalized Domination Problems. In: Paul, C., Habib, M. (eds) Graph-Theoretic Concepts in Computer Science. WG 2009. Lecture Notes in Computer Science, vol 5911. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11409-0_12
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DOI: https://doi.org/10.1007/978-3-642-11409-0_12
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