Covering graphs with few complete bipartite subgraphs

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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. © 2008 Elsevier B.V. All rights reserved.

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Fleischner, H., Mujuni, E., Paulusma, D., & Szeider, S. (2009). Covering graphs with few complete bipartite subgraphs. Theoretical Computer Science, 410(21–23), 2045–2053. https://doi.org/10.1016/j.tcs.2008.12.059

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