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

Lacroix, Mathieu; Mahjoub, Ali Ridha; Martin, Sébastien; Picouleau, Christophe (2012), On the NP-completeness of the perfect matching free subgraph problem, Theoretical Computer Science, 423, p. 25-29. http://dx.doi.org/10.1016/j.tcs.2011.12.065

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Type
Article accepté pour publication ou publié
Date
2012
Journal name
Theoretical Computer Science
Volume
423
Publisher
Elsevier
Pages
25-29
Publication identifier
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Author(s)
Lacroix, Mathieu cc
Mahjoub, Ali Ridha
Martin, Sébastien cc
Picouleau, Christophe cc
Abstract (EN)
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.
Subjects / Keywords
Bipartite graph; Matching; Structural analysis problem; NP-complete; Tripartite graph; Stable set; Blocker