Recognizing Chordal-Bipartite Probe Graphs

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 thatG 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.