  The structure of the graph defined by the interactions in a Boolean network
can determine properties of the asymptotic dynamics. For instance, considering
the asynchronous dynamics, the absence of positive cycles guarantees the
existence of a unique attractor, and the absence of negative cycles ensures
that all attractors are fixed points. In presence of multiple attractors, one
might be interested in properties that ensure that attractors are sufficiently
"isolated", that is, they can be found in separate subspaces or even trap
spaces, subspaces that are closed with respect to the dynamics. Here we
introduce notions of separability for attractors and identify corresponding
necessary conditions on the interaction graph. In particular, we show that if
the interaction graph has at most one positive cycle, or at most one negative
cycle, or if no positive cycle intersects a negative cycle, then the attractors
can be separated by subspaces. If the interaction graph has no path from a
negative to a positive cycle, then the attractors can be separated by trap
spaces. Furthermore, we study networks with interaction graphs admitting two
vertices that intersect all cycles, and show that if their attractors cannot be
separated by subspaces, then their interaction graph must contain a copy of the
complete signed digraph on two vertices, deprived of a negative loop. We thus
establish a connection between a dynamical property and a complex network
motif. The topic is far from exhausted and we conclude by stating some open
questions.
