Covering graphs with few complete bipartite subgraphs

  • Authors:

  • Herbert Fleischner;Egbert Mujuni;Daniël Paulusma;Stefan Szeider

  • Affiliations:

  • Department of Computer Science, Vienna Technical University, A-1040 Vienna, Austria;Mathematics Department, University of Dar es Salaam, PO Box 35062, Dar es Salaam, Tanzania;Department of Computer Science, Durham University, Durham DH1 3LE, United Kingdom;Department of Computer Science, Durham University, Durham DH1 3LE, United Kingdom

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2009

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Abstract

We consider computational problems on covering graphs with bicliques (complete bipartite subgraphs). Given a graph and an integer k, the biclique cover problem asks whether the edge-set of the graph can be covered with at most k bicliques; the biclique partition problem is defined similarly with the additional condition that the bicliques are required to be mutually edge-disjoint. The biclique vertex-cover problem asks whether the vertex-set of the given graph can be covered with at most k bicliques, the biclique vertex-partition problem is defined similarly with the additional condition that the bicliques are required to be mutually vertex-disjoint. All these four problems are known to be NP-complete even if the given graph is bipartite. In this paper, we investigate them in the framework of parameterized complexity: do the problems become easier if k is assumed to be small? We show that, considering k as the parameter, the first two problems are fixed-parameter tractable, while the latter two problems are not fixed-parameter tractable unless P=NP.