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 $\phi(G)$ 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 $\phi(G)\leq 2^{\tau}$, where $\tau$ is the minimum size of a feedback vertex set of $G$. In this paper, we study the maximum number $\phi_m(G)$ of fixed points in a {\em monotone} Boolean network with interaction graph $G$. We establish new upper and lower bounds on $\phi_m(G)$ that depends on the cycle structure of $G$. In addition to $\tau$, the involved parameters are the maximum number $\nu$ of vertex-disjoint cycles, and the maximum number $\nu^{*}$ of vertex-disjoint cycles verifying some additional technical conditions. We improve the classical upper bound $2^\tau$ by proving that $\phi_m(G)$ is at most the largest sub-lattice of $\{0,1\}^\tau$ without chain of size $\nu+1$, and without another forbidden-pattern of size $2\nu^{*}$. Then, we prove two optimal lower bounds: $\phi_m(G)\geq \nu+1$ and $\phi_m(G)\geq 2^{\nu^{*}}$. As a consequence, we get the following characterization: $\phi_m(G)=2^\tau$ if and only if $\nu^{*}=\tau$. As another consequence, we get that if $c$ is the maximum length of a chordless cycle of $G$ then $2^{\nu/3^c}\leq\phi_m(G)\leq 2^{c\nu}$. 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. 
Julio Aracena, Adrien Richard and Lilian Salinas
math.CO cs.DM cs.IT math.IT q-bio.MN