Fixed points and connections between positive and negative cycles in Boolean networks

We are interested in the relationships between the number fixed points in a Boolean network $f:\{0,1\}^n\to\{0,1\}^n$ and its interaction graph, which is the arc-signed digraph $G$ on $\{1,\dots,n\}$ that describes the positive and negative influences between the components of the network. A fundamental theorem of Aracena says that if $G$ has no positive (resp. negative) cycle, then $f$ has at most (resp. at least) one fixed point; the sign of a cycle being the product of the signs of its arcs. In this note, we generalize this result by taking into account the influence of connections between positive and negative cycles. In particular, we prove that if every positive (resp. negative) cycle of $G$ has an arc $a$ such that $G\setminus a$ has a non-trivial initial strongly connected component containing the terminal vertex of $a$ and only negative (resp. positive) cycles, then $f$ has at most (resp. at least) one fixed point. This is, up to our knowledge, the first generalization of Aracena's theorem where the conditions are expressed with $G$ only.