Hopcroft\u2019s Algorithm and Tree-like Automata

In order to analyze some extremal cases of Hopcroft\u2019s algorithm we deepened the relationship between combinatorial properties of circular words and the ex- ecution of the algorithm on cyclic automata associated to such words.In this paper we highlight the notion of word tree and in particular, we char- acterize the word trees for which Hopcroft\u2019s algorithm on the associated tree-like automata has a unique refinement process. Moreover, we show the relationship between the time complexity of the refinements process of the Hopcroft\u2019s algo- rithm on unary cyclic automata and binary tree-like automata. Such a result allows to exhibit a family of tree-like automata representing the worst case of the algorithm