Parameterized complexity of generalized domination problems

  • Authors:

  • Petr A. Golovach;Jan Kratochvíl;Ondřej Suchý

  • Affiliations:

  • School of Engineering and Computing Sciences, Durham University, South Road, DH1 3LE Durham, UK;Department of Applied Mathematics and Institute for Theoretical Computer Science (ITI), Charles University, 11800 Prague, Czech Republic11ITI is supported by the Ministry of Education of the Czech ...;Department of Applied Mathematics and Institute for Theoretical Computer Science (ITI), Charles University, 11800 Prague, Czech Republic11ITI is supported by the Ministry of Education of the Czech ...

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2012

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Abstract

Given two sets @s,@r of non-negative integers, a set S of vertices of a graph G is (@s,@r)-dominating if |S@?N(v)|@?@s for every vertex v@?S, and |S@?N(v)|@?@r 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 (@s,@r)-domination in this general setting. Among other results, we show that the existence of a (@s,@r)-dominating set of size k (and at most k) are W[1]-complete problems (when parameterized by k) for any pair of finite sets @s and @r. We further present results on dual parameterization by n-k, and results on certain infinite sets (in particular for @s,@r being the sets of even and odd integers).