Abstract
A graph G=(V,E) is Cprobe if V can be partitioned into two sets, probes P and non-probes N, where N is independent and new edges may be added between non-probes such that the resulting graph is in the graph class C. We say that (N,P) is a Cprobe partition for G. The Cunpartitioned probe problem consists of an input graph G and the question: Is G a C probe graph? A (k,ℓ)-partition of a graph G is a partition of its vertex set into at most k independent sets and ℓ cliques. A graph is (k,ℓ) if it has a (k,ℓ)-partition. We prove the full complexity dichotomy into NP-complete and polynomial for (k,ℓ)unpartitioned probe problems: they are NP-complete if k+ℓ≥3, and polynomial otherwise. This gives the first examples of graph classes C that can be recognized in polynomial time but whose probe graph classes are NP-complete.
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