Characterizations, probe and sandwich problems on (k,ℓ)-cographs

A cograph is a graph without an induced 3-edge path. A graph is (k,ℓ) if its vertex set can be partitioned into at most k independent sets and ℓ cliques. (k,ℓ)-cographs already have a forbidden induced subgraphs characterization, but no structural characterization is known, except for (1,1)-cographs, i.e. threshold graphs. In this paper, we present a structural characterization and a decomposition theorem for (2,1)-cographs and, consequently, for (1,2)-cographs, leading to linear time recognition algorithms for both classes. Since recognizing a graph is a very important tool to solve several other problems, besides providing these characterizations, we applied them while dealing with two closely related problems: graph sandwich problems and probe problems. graph sandwich problems for property Π (Π-sp) were introduced by Golumbic et al. as a natural generalization of recognition problems. Probe problems were defined by Zhang and can be analyzed in two versions, partitioned, which is a particular case of a sandwich problem, and non-partitioned. Our main result on probe problems shows that the probe (2,1)-cograph problem, in both versions, is polynomial-time solvable. Moreover, we also prove that, although the cograph-sp and the threshold-sp are polynomial-time solvable problems, the (2,1)-cograph-sp and the join of two thresholds-sp are NP-complete problems. As a corollary, we have that the (1,2)-cograph-sp is NP-complete as well. Using these results, we can fully classify P vs NP-complete dichotomy for the (k,ℓ) -cograph-sp, for fixed integers k,ℓ.