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On Making a Distinguished Vertex of Minimum Degree by Vertex Deletion

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Abstract

For directed and undirected graphs, we study how to make a distinguished vertex the unique minimum-(in)degree vertex through deletion of a minimum number of vertices. The corresponding NP-hard optimization problems are motivated by applications concerning control in elections and social network analysis. Continuing previous work for the directed case, we show that the problem is W[2]-hard when parameterized by the graph’s feedback arc set number, whereas it becomes fixed-parameter tractable when combining the parameters “feedback vertex set number” and “number of vertices to delete”. For the so far unstudied undirected case, we show that the problem is NP-hard and W[1]-hard when parameterized by the “number of vertices to delete”. On the positive side, we show fixed-parameter tractability for several parameterizations measuring tree-likeness. In particular, we provide a dynamic programming algorithm for graphs of bounded treewidth and a vertex-linear problem kernel with respect to the parameter “feedback edge set number”. On the contrary, we show a non-existence result concerning polynomial-size problem kernels for the combined parameter “vertex cover number and number of vertices to delete”, implying corresponding non-existence results when replacing vertex cover number by treewidth or feedback vertex set number.

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Fig. 1
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Notes

  1. Observation 1 does not hold for a feedback vertex set containing the distinguished vertex. Hence, the following approach does not transfer to this more general case.

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Acknowledgement

We are grateful to two anonymous referees of Algorithmica whose constructive feedback helped to improve the quality of our presentation.

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Authors and Affiliations

  1. Institut für Softwaretechnik und Theoretische Informatik, TU Berlin, Berlin, Germany

    Nadja Betzler, Robert Bredereck, Rolf Niedermeier & Johannes Uhlmann

  2. Department of Information and Computing Sciences, Utrecht University, Utrecht, The Netherlands

    Hans L. Bodlaender

Authors
  1. Nadja Betzler
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  2. Hans L. Bodlaender
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  3. Robert Bredereck
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  4. Rolf Niedermeier
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  5. Johannes Uhlmann
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Corresponding author

Correspondence to Robert Bredereck.

Additional information

An extended abstract of this paper appeared in Proceedings of the 37th International Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM-2011), January 22–28, 2011, Nový Smokovec, Slovakia, volume 6543 in Lecture Notes in Computer Science, pages 123–134, Springer, 2011. Compared to the conference version, the most important change (besides providing missing details) here is that we provide a concrete tree decomposition-based dynamic programming algorithm for Min-Degree Deletion parameterized by treewidth while the conference version just claimed a classification result based on monadic second-order logic.

N. Betzler and R. Bredereck were supported by the DFG, research project PAWS, NI 369/10.

J. Uhlmann was supported by the DFG, research project PABI, NI 369/7.

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Betzler, N., Bodlaender, H.L., Bredereck, R. et al. On Making a Distinguished Vertex of Minimum Degree by Vertex Deletion. Algorithmica 68, 715–738 (2014). https://doi.org/10.1007/s00453-012-9695-6

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  • DOI: https://doi.org/10.1007/s00453-012-9695-6

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