On making a distinguished vertex minimum degree by vertex deletion

  • Authors:

  • Nadja Betzler;Robert Bredereck;Rolf Niedermeier;Johannes Uhlmann

  • Affiliations:

  • Institut für Informatik, Friedrich-Schiller-Universität Jena, Jena, Germany;Institut für Softwaretechnik und Theoretische Informatik, Technische Universität Berlin, Berlin, Germany;Institut für Softwaretechnik und Theoretische Informatik, Technische Universität Berlin, Berlin, Germany;Institut für Informatik, Friedrich-Schiller-Universität Jena, Jena, Germany

  • Venue:
  • SOFSEM'11 Proceedings of the 37th international conference on Current trends in theory and practice of computer science
  • Year:
  • 2011

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Abstract

For directed and undirected graphs, we study the problem 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 "feed-back 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, including 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 nonexistence results when replacing vertex cover number by treewidth or feedback vertex set number.