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International Workshop on Graph-Theoretic Concepts in Computer Science

WG 1996: Graph-Theoretic Concepts in Computer Science pp 323–337Cite as

Homogeneous sets and domination problems

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1197))

Abstract

In this paper we consider the relation of homogeneous sets (sometimes called modules) to the domination problems r-dominating clique and connected r-dominating set by investigating homogeneous extensions of graphs.

At first, we show that homogeneous extensions of a hereditary graph class \(\mathcal{G}\)can be recognized nearly as efficiently as the graphs of \(\mathcal{G}\), itself. The algorithm is based on modular decomposition.

In the main part of this work we show, that efficient algorithms solving the r-dominating clique and the connected r-dominating set problem (and thus the Steiner tree problem) on a hereditary graph class \(\mathcal{G}\)lead to efficient algorithms on their homogeneous extensions.

Applying these results to homogeneous extensions of trees we get efficient algorithms solving these problems in linear sequential and polylogarithmic parallel time using a linear number of processors.

This work is supported by the German Research Community DFG.

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Author information

Authors and Affiliations

  1. FB Mathematik, FG Informatik, Gerhard-Mercator-Universität - GH Duisburg, D-47048, Duisburg, Germany

    Falk Nicolai

  2. FB Informatik, Lehrstuhl für Theoretische Informatik, Universität Rostock, Albert-Einstein-Str. 21, D-18051, Rostock, Germany

    Thomas Szymczak

Authors
  1. Falk Nicolai
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  2. Thomas Szymczak
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Editor information

Fabrizio d'Amore Paolo Giulio Franciosa Alberto Marchetti-Spaccamela

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© 1997 Springer-Verlag Berlin Heidelberg

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Nicolai, F., Szymczak, T. (1997). Homogeneous sets and domination problems. In: d'Amore, F., Franciosa, P.G., Marchetti-Spaccamela, A. (eds) Graph-Theoretic Concepts in Computer Science. WG 1996. Lecture Notes in Computer Science, vol 1197. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-62559-3_26

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  • DOI: https://doi.org/10.1007/3-540-62559-3_26

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-62559-9

  • Online ISBN: 978-3-540-68072-7

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