arXiv2026
The central object in this work is the Boolean network, a function $f:\{0,1\}^V \to \{0,1\}^V$. Often identified with $f$ is its dynamics $D_f$, the digraph with exactly $\{x \to y : f(x)=y\}$ as its arcs. The interaction graph of $f$ is the digraph $G$ on $V$ which can be thought of as representing the dependencies of the components of $f$ on one another. On the other hand, model-checking is a major field of study in graph algorithms and complexity. We consider problems at the intersection of these paradigms: for a given property $ϕ$, we say that a digraph $G$ is $ϕ$-dynamic if $G$ is the interaction graph of some Boolean network $f$ such that $D_f \models ϕ$. The corresponding decision problem $ϕ$-DYNAMIC asks whether $G$ is $ϕ$-dynamic. Problems of this form are well-studied in the Boolean networks community. For example, it is a well-known result that, letting $ϕ$ be 'No fixed points' (i.e. $\nexists x:x\to x$), $ϕ$-DYNAMIC is equivalent to the structural digraph problem NON-ACYCLIC, which is solvable in linear time. We establish, for the first time, properties $ϕ$ for which $ϕ$-DYNAMIC is computationally hard. These include the 'Unique fixed point' property $\exists!x:x\to x$, and a range of properties concerning the periodic points of $f$ (i.e. vertices of $D_f$ which lie on a cycle). More widely, we exhibit an infinite family of natural properties $ϕ$ such that $ϕ$-DYNAMIC is $\mathrm{NEXPTIME}$-complete, which we call PERIOD PIECE. As a special case, these include all properties of the form 'Unique cycle of length $k$' and 'Exactly $k$ fixed points' for any $k \ge 1$. On the other hand, we show that if the input digraph has bounded in-degree, then for those same problems $ϕ$-DYNAMIC collapses down to $\mathrm{NP}^{\mathrm{NP}}$, the second level of the polynomial hierarchy, though it remains $\mathrm{NP}^{\mathrm{NP}}$-complete.