### Abstract

Given an undirected graph with weights on its vertices, the k most vital nodes independent set (k most vital nodes vertex cover) problem consists of determining a set of k vertices whose removal results in the greatest decrease in the maximum weight of independent sets (minimum weight of vertex covers, respectively). We also consider the complementary problems, minimum node blocker independent set (minimum node blocker vertex cover) that consists of removing a subset of vertices of minimum size such that the maximum weight of independent sets (minimum weight of vertex covers, respectively) in the remaining graph is at most a specified value. We show that these problems are NP-hard on bipartite graphs but polynomial-time solvable on unweighted bipartite graphs. Furthermore, these problems are polynomial also on cographs and graphs of bounded treewidth. Results on the non-existence of ptas are presented, too.

Original language | English |
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Pages (from-to) | 1933-1946 |

Number of pages | 14 |

Journal | Discrete Applied Mathematics |

Volume | 159 |

Issue number | 17 |

DOIs | |

Publication status | Published - Oct 28 2011 |

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### Keywords

- Bipartite graph
- Bounded treewidth
- Cograph
- Independent set
- Most vital vertices
- NP-hard
- Time complexity
- Vertex cover

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Discrete Applied Mathematics*,

*159*(17), 1933-1946. https://doi.org/10.1016/j.dam.2011.06.023