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We consider the problem of minimizing the makespan in open shop scheduling. The decision problem whether a given sequence in open shop scheduling is irreducible has already been considered, however, has not been solved yet. A sequence is an acyclic orientation of the Hamming graph K n × K m . Irreducible sequences in open shop are the local optimal elements. We present two variants of algorithms based on the specific properties of the H-comparability graph. The first is polynomial whereas the second is exponential. The irreducibility is co-NP. The stated properties argue whether it belongs to P . The complexity status of the considered decision problem is updated.

A comparability graph is a graph whose edges can be oriented transitively. Given a comparability graph G = (V, E) and an arbitrary edgê e ∈ E we explore the question whether the graph G −ê, obtained by removing the undirected edgeê, is a comparability graph as well. We define a new substructure of implication classes and present a complete mathematical characterization of all those edges.