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New Polynomial Cases of the Weighted Efficient Domination Problem

  • Conference paper
Mathematical Foundations of Computer Science 2013 (MFCS 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8087))

Abstract

Let G be a finite undirected graph. A vertex dominates itself and its neighbors in G. A vertex set D is an efficient dominating set (e.d. for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient Domination (ED) problem, which asks for the existence of an e.d. in G, is known to be \(\mathbb{NP}\)-complete even for very restricted graph classes.

In particular, the ED problem remains \(\mathbb{NP}\)-complete for 2P 3-free graphs and thus for P 7-free graphs. We show that the weighted version of the problem (abbreviated WED) is solvable in polynomial time on various subclasses of P 7-free graphs, including (P 2 + P 4)-free graphs, P 5-free graphs and other classes.

Furthermore, we show that a minimum weight e.d. consisting only of vertices of degree at most 2 (if one exists) can be found in polynomial time. This contrasts with our \(\mathbb{NP}\)-completeness result for the ED problem on planar bipartite graphs with maximum degree 3.

This work is supported in part by the Slovenian Research Agency (research program P1–0285 and research projects J1–4010, J1–4021, MU-PROM/2012–022 and N1–0011: GReGAS, supported in part by the European Science Foundation).

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Authors and Affiliations

  1. Institut für Informatik, Universität Rostock, D-18051, Rostock, Germany

    Andreas Brandstädt & Ragnar Nevries

  2. UP IAM and UP FAMNIT, University of Primorska, SI6000, Koper, Slovenia

    Martin Milanič

Authors
  1. Andreas Brandstädt
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  2. Martin Milanič
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  3. Ragnar Nevries
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Editors and Affiliations

  1. Institute for Science and Technology, 3400, Klosterneuburg, Austria

    Krishnendu Chatterjee

  2. Charles University, 11800, Prague, Czech Republic

    Jirí Sgall

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Brandstädt, A., Milanič, M., Nevries, R. (2013). New Polynomial Cases of the Weighted Efficient Domination Problem. In: Chatterjee, K., Sgall, J. (eds) Mathematical Foundations of Computer Science 2013. MFCS 2013. Lecture Notes in Computer Science, vol 8087. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40313-2_19

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  • DOI: https://doi.org/10.1007/978-3-642-40313-2_19

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-40312-5

  • Online ISBN: 978-3-642-40313-2

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