On the NP-completeness of the perfect matching free subgraph problem

Authors:
Mathieu Lacroix;A. Ridha Mahjoub;Sébastien Martin;Christophe Picouleau
Affiliations:
Université Paris-Dauphine, LAMSADE, CNRS UMR 7243, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France;Université Paris-Dauphine, LAMSADE, CNRS UMR 7243, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France;Université Paris-Dauphine, LAMSADE, CNRS UMR 7243, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France;Conservatoire National des Arts et Métiers, CEDRIC, 292 Rue St Martin, 75141 Paris, France
Venue:
Theoretical Computer Science
Year:
2012

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Abstract

Given a bipartite graph G=(U@?V,E) such that |U|=|V| and every edge is labelled true or false or both, the perfect matching free subgraph problem is to determine whether or not there exists a subgraph of G containing, for each node u of U, either all the edges labelled true or all the edges labelled false incident to u, and which does not contain a perfect matching. This problem arises in the structural analysis of differential-algebraic systems. The purpose of this paper is to show that this problem is NP-complete. We show that the problem is equivalent to the stable set problem in a restricted case of tripartite graphs. Then we show that the latter remains NP-complete in that case. We also prove the NP-completeness of the related minimum blocker problem in bipartite graphs with perfect matching.