Branch and recharge: 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, Bergen, Norway;Department of Informatics, University of Bergen, Bergen, Norway;Department of Applied Mathematics, and Institute for Theoretical Computer Science, Charles University, Praha, Czech Republic;LITA, Université Paul Verlaine, Metz Cedex, France;LITA, Université Paul Verlaine, Metz Cedex, France

  • Venue:
  • WADS'07 Proceedings of the 10th international conference on Algorithms and Data Structures
  • Year:
  • 2007

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Abstract

Let σ and δ be two sets of nonnegative integers. A vertex subset S ⊆ V of an undirected graph G = (V,E) is called a (σ, δ)-dominating set of G if |N(v)∩S| ∈ σ for all v ∈ S and |N(v)∩S| ∈ δ for all v ∈ V \S. This notion introduced by Telle generalizes many dominationtype graph invariants. For many particular choices of σ and δ it is NP complete to decide whether an input graph has a (σ, δ)-dominating set. We show a general algorithm enumerating all (σ, δ)-dominating sets of an input graph G in time O*(cn) for some c o(2n). Our algorithm straightforwardly implies a non trivial upper bound cn with c Finally, we also present algorithms to find maximum and minimum ({p}, {q})-dominating sets and to count the ({p}, {q})-dominating sets of a graph in time O*(2n/2).