Local negative circuits and fixed points in Boolean networks

Original · EN

To each Boolean function F from 0,1ⁿ to itself and each point x in 0,1ⁿ, we associate the signed directed graph GF(x) of order n that contains a positive (resp. negative) arc from j to i if the partial derivative of fᵢ with respect of xⱼ is positive (resp. negative) at point x. We then focus on the following open problem: Is the absence of a negative circuit in GF(x) for all x in 0,1ⁿ a sufficient condition for F to have at least one fixed point? As main result, we settle this problem under the additional condition that, for all x in 0,1ⁿ, the out-degree of each vertex of GF(x) is at most one.