Recognizing Chordal-Bipartite Probe Graphs

Anne Berry (1) , Elad Cohen (2) , Martin C. Golumbic (2) , Marina Lipshteyn (2) , Nicolas Pinet (1) , Alain Sigayret (1) , Michal Stern (2)

1 LIMOS - Laboratoire d'Informatique, de Modélisation et d'optimisation des Systèmes
2 Caesarea Rothschild Institute and Department of Computer Science

A graph G is chordal-bipartite probe if its vertices can be partitioned into two sets P (probes) and N (non-probes) where N is a stable set and such that G can be extended to a chordal-bipartite graph by adding edges between non-probes. A bipartite graph is called chordal-bipartite if it contains no chordless cycle of length strictly greater than 5. Such probe/non-probe completion problems have been studied previously on other families of graphs, such as interval graphs and chordal graphs. In this paper, we give a characterization of chordal-bipartite probe graphs, in the case of a fixed given partition of the vertices into probes and nonprobes. Our results are obtained by solving first the more general case without assuming that N is a stable set, and then this can be applied to the more specific case. Our characterization uses an edge elimination ordering which also implies a polynomial time recognition algorithm for the class. This research was conducted in the context of a France-Israel Binational project, while the French team visited Haifa in March 2007.

Mots clés: Chordal-bipartite graphs; probe graphs; elimination schemes

Domaines: Recherche opérationnelle [math.OC]

Anne Berry, Elad Cohen, Martin C. Golumbic, Marina Lipshteyn, Nicolas Pinet, et al.. Recognizing Chordal-Bipartite Probe Graphs. 2007. ⟨hal-00678308⟩