Probe threshold and probe trivially perfect graphs

An undirected graph G=(V,E) is a probe C graph if its vertex set can be partitioned into two sets, N (nonprobes) and P (probes) where N is independent and there exists E′⊆N×N such that G′=(V,E∪E′) is a C graph. In this article we investigate probe threshold and probe trivially perfect graphs and characterise them in terms of certain 2-Sat formulas and in other ways. For the case when the partition into probes and nonprobes is given, we give characterisations by forbidden induced subgraphs, linear recognition algorithms (in the case of probe threshold graphs it is based on the degree sequence of the graph), and linear algorithms to find a set E′ of minimum size. Furthermore, we give linear time recognition algorithms for both classes and a characterisation by forbidden subgraphs for probe threshold graphs when the partition (P,N) is not given.