Elsevier

Discrete Applied Mathematics

Volume 159, Issue 17, 28 October 2011, Pages 1933-1946
Discrete Applied Mathematics

The most vital nodes with respect to independent set and vertex cover

https://doi.org/10.1016/j.dam.2011.06.023Get rights and content
Under an Elsevier user license
open archive

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.

Highlights

k most vital nodes independent set (vertex cover) and their dual are studied.► They are proved NP-hard on bipartite graphs but polynomial on unweighted bipartite graphs.► They are proved polynomial on trees, cographs and graphs of bounded treewidth.►k most vital versions have no ptas.

Keywords

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

Cited by (0)