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Domination Above r-Independence: Does Sparseness Help?

Authors Carl Einarson, Felix Reidl

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ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Dominating Set
  • Above Guarantee
  • Kernel
  • Bounded Expansion
  • Nowhere Dense

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Abstract

Inspired by the potential of improving tractability via gap- or above-guarantee parametrisations, we investigate the complexity of Dominating Set when given a suitable lower-bound witness. Concretely, we consider being provided with a maximal r-independent set X (a set in which all vertices have pairwise distance at least r+1) along the input graph G which, for r >= 2, lower-bounds the minimum size of any dominating set of G. In the spirit of gap-parameters, we consider a parametrisation by the size of the "residual" set R := V(G) \ N[X].
Our work aims to answer two questions: How does the constant r affect the tractability of the problem and does the restriction to sparse graph classes help here? For the base case r = 2, we find that the problem is paraNP-complete even in apex- and bounded-degree graphs. For r = 3, the problem is W[2]-hard for general graphs but in FPT for nowhere dense classes and it admits a linear kernel for bounded expansion classes. For r >= 4, the parametrisation becomes essentially equivalent to the natural parameter, the size of the dominating set.

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Carl Einarson and Felix Reidl. Domination Above r-Independence: Does Sparseness Help?. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 40:1-40:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.MFCS.2019.40

Author Details

Carl Einarson
  • Royal Holloway, University of London, UK
Felix Reidl
  • Birkbeck, University of London, UK

Acknowledgements

We thank our anonymous reviewer for helpfully pointing out how to achieve a linear kernel in Theorem 14.

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