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We study inductive and second-order definability on finite structures with successor and relate these notions to complexity theory. We introduce the dimension d of an inductive definition, directly related to the Time complexity, representing the number of variables necessary to carry an induction. We will prove the following: Connectivity is not uniformly unary &Sgr;11 definable on graphs with successor, generalizing a similar result of Fagin on graphs without successor. We will mention that similarly, Hamiltonicity is neither unary-&Sgr;11 nor unary-&pgr;11 on graphs with successor. These results show that disconnectivity is not inductive of dimension 1 although it is inductive of dimension 2, and that Hamiltonicity is neither inductive nor co-inductive of dimension 1 on graphs with successor.