Never minimal automata and the rainbow bipartite subgraph problem

Emanuele Rodaro and Pedro V. Silva

Abstract. 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.

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