On Acyclic Orientations and Sequential Dynamical Systems

Abstract

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 (Fi,Y: F2n→F2n)i, and (c) a permutation π∈Sn. The function Fi,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 π represents a Y-vertex ordering according to which the functions Fi,Y are applied. By composing the functions Fi,Y in the order given by π we obtain the sequential dynamical system (SDS):[FY,π]=Fπ(n),Y∘⋯∘Fπ(1),Y: F2n→F2n.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 (Fi,Y). Second, we analyze the structure of a certain class of fixed-point-free SDSs.