Sort and Search: Exact algorithms for generalized domination

  • Authors:

  • Fedor V. Fomin;Petr A. Golovach;Jan Kratochvíl;Dieter Kratsch;Mathieu Liedloff

  • Affiliations:

  • Department of Informatics, University of Bergen, 5020 Bergen, Norway;Department of Informatics, University of Bergen, 5020 Bergen, Norway;Department of Applied Mathematics, and Institute for Theoretical Computer Science, Charles University, Malostranské nám. 25, 118 00 Praha 1, Czech Republic;Laboratoire d'Informatique Théorique et Appliquée, Université Paul Verlaine - Metz, 57045 Metz Cedex 01, France;Laboratoire d'Informatique Fondamentale d'Orléans, Université d'Orléans, 45067 Orléans Cedex 2, France

  • Venue:
  • Information Processing Letters
  • Year:
  • 2009

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Abstract

In 1994, Telle introduced the following notion of domination, which generalizes many domination-type graph invariants. Let @s and @r be two sets of non-negative integers. A vertex subset S@?V of an undirected graph G=(V,E) is called a (@s,@r)-dominating set of G if |N(v)@?S|@?@s for all v@?S and |N(v)@?S|@?@r for all v@?V@?S. In this paper, we prove that decision, optimization, and counting variants of ({p},{q})-domination are solvable in time 2^|^V^|^/^2@?|V|^O^(^1^). We also show how to extend these results for infinite @s={p+m@?@?:@?@?N"0} and @r={q+m@?@?:@?@?N"0}. For the case |@s|+|@r|=3, these problems can be solved in time 3^|^V^|^/^2@?|V|^O^(^1^), and similarly to the case |@s|=|@r|=1 it is possible to extend the algorithm for some infinite sets.