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On the feedback vertex set problem for a planar graph

Über das Feedback-Vertex-Set-Problem für planare Graphen

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Abstract

An algorithm solving the feedback-vertex-set problem for planar digraphs is described. In particular, planar graphs with a certain additional condition are considered as they arise from solving systems of linear equations obtained from convection-dominated flow problems. The proposed algorithm requires a computational work linear in the size of the graph. Furthermore, a side-product is a decomposition of the graph into subsets that are of interest for block-Gauß-Seidel smoothers in multi-grid methods.

Zusammenfassung

Es wird ein Algorithmus zur Lösung des Feedback-Vertex-Set-Problems for planare gerichtete Graphen beschrieben. Eine zusätzliche Bedingung wird zugrundegelegt, die sich infolge der Herkunft des Problems aus der Lösung der linearen Gleichungssysteme für konvektionsdominierte Strömungsaufgaben ergibt. Der vorgeschlagene Algorithmus erfordert einen zur Größe des Graphen proportionalen Aufwand. Ferner ergibt sich als Nebenprodukt eine Zerlegung des Graphen in Teilgraphen, die eine Blockaufteilung induzieren und für Block-Gauß-Seidel-Glätter in Mehrgitterverfahren interessant sind.

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References

  1. Brandstädt, A.: The computational complexity of feedback vertex set, Hamiltonian circuit, dominating set, Steiner tree, and bandwidth on special perfect graphs. J. Inf. Process. Cybern. EIK23, 471–477 (1987).

    Google Scholar 

  2. Brandstädt, A., Kratsch, D.: On domination problems for permutation and other graphs. Theor. Comp. Sci.54, 181–198 (1987).

    Article  MATH  Google Scholar 

  3. Hackbusch, W.: Multi-grid methods and applications. Berlin, Heidelberg, New York, Tokyo: Springer, 1985.

    MATH  Google Scholar 

  4. Hackbusch, W.: Elliptic differential equations. Berlin, Heidelberg, New York, Tokyo: Springer, 1992.

    MATH  Google Scholar 

  5. Hackbusch, W.: Iterative solution of large sparse systems of equations. Berlin, Heidelberg, New York, Tokyo: Springer 1994. Iterative Lösung großer schwachbesetzter Gleichungssysteme, 2. deutsche Auflage. Stuttgart: Teubner, 1993.

    MATH  Google Scholar 

  6. Monien, B., Schulz, R.: Four approximation algorithms for the feedback vertex set problem. Bericht Nr 9, Paderborn 1981 — Proc. of the 7th Conf. on Graph Theoretical Concepts of Comput. Sci., pp. 315–325. München: Hanser, 1981.

    Google Scholar 

  7. Reinelt, G.: The linear ordering problem: algorithms and applications. Research and Exposition in Mathematics8. Berlin: Heldermann, 1985.

    MATH  Google Scholar 

  8. Speckenmeyer, E.: Bounds on feedback vertex sets of undirected cubic graphs. Coll. Math. Soc. Janos Bolyai42, 719–729 (1983).

    Google Scholar 

  9. Stamm, H.: Graph-theoretical concepts in computer science. Lect. Notes Compt. Sci.484, 79–89 (1992).

    Google Scholar 

  10. Wittum, G.: Filternde Zerlegungen: Schnelle Löser für große Gleichungssysteme (Filtering decomposition, fast solvers for large systems of equations). Teubner Skripten zur Numerik. Stuttgart: Teubner, 1992.

    Google Scholar 

  11. Hackbusch, W., Probst, Th.: Downwind Gauß-Seidel smoothing for convection dominated problems. Numer. Lin. Alg. Appl. (to appear).

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  1. Praktische Mathematik, Christian-Albrechts-Universität zu Kiel, D-24098, Kiel, Germany

    W. Hackbusch

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  1. W. Hackbusch
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Hackbusch, W. On the feedback vertex set problem for a planar graph. Computing 58, 129–155 (1997). https://doi.org/10.1007/BF02684436

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  • DOI: https://doi.org/10.1007/BF02684436

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