On the complexity of probe and sandwich problems for generalized threshold graphs

Abstract

A cograph is a graph without induced P4. A graph G is (k, ℓ) if its vertex set can be partitioned into at most k independent sets and ℓ cliques. Threshold graphs are cographs-(1, 1). We proved recently that cographs-(2, 1) are their generalization and, as threshold graphs, they can be recognized in linear time. graph sandwich problems for property Π (Π-sp) were defined by Golumbic et al. as a natural generalization of recognition problems. partitioned probe problems are particular cases of graph sandwich problems. In this paper we show that, similarly to probe threshold graphs and probe cographs, probe cographs-(2, 1) and probe join of two thresholds are recognizable in polynomial time. In contrast, although cograph-sp and threshold-sp are polynomially solvable problems, we prove that cograph-(2, 1)-sp and join of two thresholds -sp are NP-complete problems.