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We study a class of discrete dynamical systems that consists of the following data: (a) a finite (labeled) graph Y with vertex set {1,...,n}, where each vertex has a binary state, (b) a vertex labeled multi-set of functions (F"i","Y: F"2^n-F"2^n)"i, and (c) a permutation @p@?S"n. The function F"i","Y updates the binary state of vertex i as a function of the states of vertex i and its Y-neighbors and leaves the states of all other vertices fixed. The permutation @p represents a Y-vertex ordering according to which the functions F"i","Y are applied. By composing the functions F"i","Y in the order given by @p we obtain the sequential dynamical system (SDS):[F"Y,@p]=F"@p"("n")","Y@?...@?F"@p"("1")","Y: F"2^n-F"2^n.In this paper we first establish a sharp, combinatorial upper bound on the number of non-equivalent SDSs for fixed graph Y and multi-set of functions (F"i","Y). Second, we analyze the structure of a certain class of fixed-point-free SDSs.