On the NP-Completeness of the Perfect Perfect Matching Free Subgraph Problem

Mathieu Lacroix (1, 2) , Ridha Mahjoub (2) , Sébastien Martin (2) , Christophe Picouleau (3)

1 LIMOS - Laboratoire d'Informatique, de Modélisation et d'optimisation des Systèmes
2 LAMSADE - Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision
3 CEDRIC - Centre d'études et de recherche en informatique et communications

Given a bipartite graph G = (U υ 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.

Mots clés: Bipartite graph; matchιng; blocker; stable set; tripartite graph; NP-complete; structural analysis problem

Domaines: Recherche opérationnelle [math.OC]; Mathématique discrète [cs.DM]; Complexité [cs.CC]

Mathieu Lacroix, Ridha Mahjoub, Sébastien Martin, Christophe Picouleau. On the NP-Completeness of the Perfect Perfect Matching Free Subgraph Problem. 2011. ⟨hal-00875532⟩