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Efficient Dominating and Edge Dominating Sets for Graphs and Hypergraphs

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Algorithms and Computation (ISAAC 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7676))

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Abstract

Let G = (V,E) be a graph. A vertex dominates itself and all its neighbors, i.e., every vertex v ∈ V dominates its closed neighborhood N[v]. A vertex set D in G is an efficient dominating (e.d.) set for G if for every vertex v ∈ V, there is exactly one d ∈ D dominating v. An edge set M ⊆ E is an efficient edge dominating (e.e.d.) set for G if it is an efficient dominating set in the line graph L(G) of G. The ED problem (EED problem, respectively) asks for the existence of an e.d. set (e.e.d. set, respectively) in the given graph.

We give a unified framework for investigating the complexity of these problems on various classes of graphs. In particular, we solve some open problems and give linear time algorithms for ED and EED on dually chordal graphs.

We extend the two problems to hypergraphs and show that ED remains \(\mathbb{NP}\)-complete on α-acyclic hypergraphs, and is solvable in polynomial time on hypertrees, while EED is polynomial on α-acyclic hypergraphs and \(\mathbb{NP}\)-complete on hypertrees.

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Authors and Affiliations

  1. Institut für Informatik, Universität Rostock, Germany

    Andreas Brandstädt & Arne Leitert

  2. Institut für Optimierung und Operations Research, Universität Ulm, Germany

    Dieter Rautenbach

Authors
  1. Andreas Brandstädt
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  2. Arne Leitert
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  3. Dieter Rautenbach
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Editor information

Editors and Affiliations

  1. Department of Computer Science and Information Engineering, National Taiwan University, 1 Roosevelt Road, Section 4, 106, Taipei, Taiwan

    Kun-Mao Chao

  2. Institute of Information Science, Academia Sinica, 128 Academia Road, Section 2, 115, Taipei, Taiwan

    Tsan-sheng Hsu

  3. Department of Computer Science and Engineering, National Chung Hsing University, 250 Kuo Kuang Road, 402, Taichung, Taiwan

    Der-Tsai Lee

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Brandstädt, A., Leitert, A., Rautenbach, D. (2012). Efficient Dominating and Edge Dominating Sets for Graphs and Hypergraphs. In: Chao, KM., Hsu, Ts., Lee, DT. (eds) Algorithms and Computation. ISAAC 2012. Lecture Notes in Computer Science, vol 7676. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35261-4_30

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  • DOI: https://doi.org/10.1007/978-3-642-35261-4_30

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-35260-7

  • Online ISBN: 978-3-642-35261-4

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