Number of fixed points and disjoint cycles in monotone Boolean networks

Given a digraph $G$, a lot of attention has been deserved on the maximum number $φ(G)$ of fixed points in a Boolean network $f:\{0,1\}^n\to\{0,1\}^n$ with $G$ as interaction graph. In particular, a central problem in network coding consists in studying the optimality of the classical upper bound $φ(G)\leq 2^τ$, where $τ$ is the minimum size of a feedback vertex set of $G$. In this paper, we study the maximum number $φ_m(G)$ of fixed points in a {\em monotone} Boolean network with interaction graph $G$. We establish new upper and lower bounds on $φ_m(G)$ that depends on the cycle structure of $G$. In addition to $τ$, the involved parameters are the maximum number $ν$ of vertex-disjoint cycles, and the maximum number $ν^{*}$ of vertex-disjoint cycles verifying some additional technical conditions. We improve the classical upper bound $2^τ$ by proving that $φ_m(G)$ is at most the largest sub-lattice of $\{0,1\}^τ$ without chain of size $ν+1$, and without another forbidden-pattern of size $2ν^{*}$. Then, we prove two optimal lower bounds: $φ_m(G)\geq ν+1$ and $φ_m(G)\geq 2^{ν^{*}}$. As a consequence, we get the following characterization: $φ_m(G)=2^τ$ if and only if $ν^{*}=τ$. As another consequence, we get that if $c$ is the maximum length of a chordless cycle of $G$ then $2^{ν/3^c}\leqφ_m(G)\leq 2^{cν}$. Finally, with the technics introduced, we establish an upper bound on the number of fixed points of any Boolean network according to its signed interaction graph.