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The Complexity of König Subgraph Problems and Above-Guarantee Vertex Cover

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Abstract

A graph is König-Egerváry if the size of a minimum vertex cover equals that of a maximum matching in the graph. These graphs have been studied extensively from a graph-theoretic point of view. In this paper, we introduce and study the algorithmic complexity of finding König-Egerváry subgraphs of a given graph. In particular, given a graph G and a nonnegative integer k, we are interested in the following questions:

  1. 1.

    does there exist a set of k vertices (edges) whose deletion makes the graph König-Egerváry?

  2. 2.

    does there exist a set of k vertices (edges) that induce a König-Egerváry subgraph?

We show that these problems are NP-complete and study their complexity from the points of view of approximation and parameterized complexity. Towards this end, we first study the algorithmic complexity of Above Guarantee Vertex Cover, where one is interested in minimizing the additional number of vertices needed beyond the maximum matching size for the vertex cover. Further, while studying the parameterized complexity of the problem of deleting k vertices to obtain a König-Egerváry graph, we show a number of interesting structural results on matchings and vertex covers which could be useful in other contexts.

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Authors and Affiliations

  1. The Institute of Mathematical Sciences, Chennai, India

    Venkatesh Raman, Saket Saurabh & C. R. Subramanian

  2. Department of Mathematics, The Indian Institute of Technology, Chennai, India

    Sounaka Mishra

  3. Department of Computer Science, RWTH Aachen University, 52056, Aachen, Germany

    Somnath Sikdar

Authors
  1. Sounaka Mishra
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  2. Venkatesh Raman
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  3. Saket Saurabh
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  4. Somnath Sikdar
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  5. C. R. Subramanian
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Corresponding author

Correspondence to Saket Saurabh.

Additional information

Preliminary versions of this paper appeared in the proceedings of the 18th and 19th International Symposium on Algorithms and Computation (ISAAC 2007 and ISAAC 2008).

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Mishra, S., Raman, V., Saurabh, S. et al. The Complexity of König Subgraph Problems and Above-Guarantee Vertex Cover. Algorithmica 61, 857–881 (2011). https://doi.org/10.1007/s00453-010-9412-2

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  • DOI: https://doi.org/10.1007/s00453-010-9412-2

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