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

Authors:
Cristina Bazgan;Sonia Toubaline;Zsolt Tuza
Affiliations:
Université Paris-Dauphine, LAMSADE, Place du Marechal de Lattre de Tassigny, 75775 Paris Cedex 16, France;Université Paris-Dauphine, LAMSADE, Place du Marechal de Lattre de Tassigny, 75775 Paris Cedex 16, France;Computer and Automation Institute, Hungarian Academy of Sciences, H-1111 Budapest, Kende u. 13-17, Hungary and Department of Computer Science and Systems Technology, University of Pannonia, H-8200 ...
Venue:
Discrete Applied Mathematics
Year:
2011

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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.