Probe split graphs

An undirected graph G=(V,E) is a probe split graph if its vertex set can be partitioned into two sets, N (non-probes) and P (probes) where N is independent and there exists E' N× N such that G'=(V,E∪ E') is a split graph. Recently Chang et al. gave an O(V 4 (V+E)) time recognition algorithm for probe split graphs. In this article we give O(V 2 +VE) time recognition algorithms and characterisations by forbidden induced subgraphs both for the case when the partition into probes and non-probes is given, and when it is not given.