# Local negative circuits and fixed points in Boolean networks - Computer Science > Discrete Mathematics

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Abstract: To each Boolean function F from {0,1}^n to itself and each point x in{0,1}^n, we associate the signed directed graph G Fx of order n that containsa positive resp. negative arc from j to i if the partial derivative of f iwith respect of x j is positive resp. negative at point x. We then focus onthe following open problem: Is the absence of a negative circuit in G Fx forall x in {0,1}^n a sufficient condition for F to have at least one fixed point?As main result, we settle this problem under the additional condition that, forall x in {0,1}^n, the out-degree of each vertex of G Fx is at most one.

Author: ** Adrien Richard**

Source: https://arxiv.org/