Never Minimal Automata and the Rainbow Bipartite Subgraph Problem

Never minimal automata, introduced in [4], are strongly connected automata which are not minimal for any choice of their final states. In [4] the authors raise the question whether recognizing such automata is a polynomial time task or not. In this paper, we show that the complement of this problem is equivalent to the problem of checking whether or not in an edge-colored graph there is a bipartite subgraph whose edges are colored using all the colors. We prove that this graph theoretic problem is NP-complete, showing that checking the property of never-minimality is unlikely a polynomial time task.