The problem of recognizing whether a subset of attributes is a premise of a minimal cover of functional dependencies of a relation is shown to be coNP-complete. The complexity of some related decision, enumerating, and sampling problems on functional dependencies, FCA implications, and closed sets of attributes is discussed.
Given a set X, a König graph G for X is a graph with the following property: for every induced subgraph H of G, the maximum number of vertex-disjoint induced subgraphs from X in H is equal to the minimum number of vertices whose deletion from H results in a graph containing no graph in X as an induced subgraph. The purpose of this paper is to characterize all König graphs for X, where X has only the 3-path or X consists of the 3-path and 3-cycle. We give also polynomial-time algorithms for the recognition of König graphs for the 3-path and for finding the corresponding packing and cover numbers in graphs of this type.