, Volume 68, Issue 3, pp 715–738 | Cite as

On Making a Distinguished Vertex of Minimum Degree by Vertex Deletion

  • Nadja Betzler
  • Hans L. Bodlaender
  • Robert Bredereck
  • Rolf Niedermeier
  • Johannes Uhlmann


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.


Graph modification problem Parameterized complexity Treewidth 



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


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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Nadja Betzler
    • 1
  • Hans L. Bodlaender
    • 2
  • Robert Bredereck
    • 1
  • Rolf Niedermeier
    • 1
  • Johannes Uhlmann
    • 1
  1. 1.Institut für Softwaretechnik und Theoretische InformatikTU BerlinBerlinGermany
  2. 2.Department of Information and Computing SciencesUtrecht UniversityUtrechtThe Netherlands

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