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} → {0, 1} 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) ≤ 2 , where τ is the minimum size of a feedback vertex set of G. In this paper, we study the maximum number φm(G) of fixed points in a monotone Boolean network with interaction graph G. We establish new upper and lower bounds on φm(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 vertexdisjoint cycles verifying some additional technical conditions. We improve the classical upper bound 2 by proving that φm(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: φm(G) ≥ ν + 1 and φm(G) ≥ 2 ∗ . As a consequence, we get the following characterization: φm(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 c ≤ φm(G) ≤ 2. 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.