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We study the parameterized complexity of two classic problems on directedgraphs: Hamiltonian Cycle and its generalization {\sc Longest Cycle}. Since2008, it is known that Hamiltonian Cycle is W[1]-hard when parameterized bydirected treewidth [Lampis et al., ISSAC'08]. By now, the question of whetherit is FPT parameterized by the directed feedback vertex set (DFVS) number hasbecome a longstanding open problem. In particular, the DFVS number is thelargest natural directed width measure studied in the literature. In thispaper, we provide a negative answer to the question, showing that even for theDFVS number, the problem remains W[1]-hard. As a consequence, we also obtainthat Longest Cycle is W[1]-hard on directed graphs when parameterizedmultiplicatively above girth, in contrast to the undirected case. This resolvesan open question posed by Fomin et al. [ACM ToCT'21] and Gutin and Mnich[arXiv:2207.12278]. Our hardness results apply to the path versions of theproblems as well. On the positive side, we show that Longest Path parameterizedmultiplicatively above girth} belongs to the class XP.

Finding Long Directed Cycles Is Hard Even When DFVS Is Small Or Girth Is Large