Probe threshold and probe trivially perfect graphs

7 An undirected graph G = (V,E) is a probe C graph if its vertex set can be partitioned 8 into two sets, N (non-probes) and P (probes) where N is independent and there 9 exists E′ ⊆ N × N such that G′ = (V,E ∪ E′) is a C graph. In this article we 10 investigate probe threshold and probe trivially perfect graphs and characterise them 11 in terms of certain 2-Sat formulas and in other ways. For the case when the partition 12 into probes and non-probes is given, we give characterisations by forbidden induced 13 subgraphs, linear recognition algorithms (in the case of probe threshold graphs it is 14 based on the degree sequence of the graph), and linear algorithms to find a set E′ 15 of minimum size. Furthermore, we give linear time recognition algorithms for both 16 classes and a characterisation by forbidden subgraphs for probe threshold graphs 17 when the partition (P,N) is not given. 18