Number of Fixed Points and Disjoint Cycles in Monotone Boolean Networks - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue SIAM Journal on Discrete Mathematics Année : 2017

Number of Fixed Points and Disjoint Cycles in Monotone Boolean Networks

Résumé

Given a digraph G, a lot of attention has been deserven on the maximum number φ(G) of fixed points in a Boolean network f : {0, 1} n → {0, 1} n with G as interaction graph. In particular, a central problem in network coding consists in studying the optimality of the feedback bound φ(G) ≤ 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 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 feedback 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 + 2, and without another forbidden pattern described by two disjoint antichains of size ν * + 1. Then, we prove two optimal lower bounds: φ m (G) ≥ ν + 1 and φ m (G) ≥ 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 ≤ φ m (G) ≤ 2 cν. Finally, with the techniques introduced, we establish an upper bound on the number of fixed points of any Boolean network according to its signed interaction graph.
Fichier principal
Vignette du fichier
2017_04_12_maxFP_Monotonous_R1.pdf (409.41 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-01630477 , version 1 (07-11-2017)

Identifiants

Citer

Julio Aracena, Adrien Richard, Lilian Salinas. Number of Fixed Points and Disjoint Cycles in Monotone Boolean Networks. SIAM Journal on Discrete Mathematics, 2017, 31 (3), pp.1702 - 1725. ⟨10.1137/16M1060868⟩. ⟨hal-01630477⟩
63 Consultations
150 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More