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The structure of the graph defined by the interactions in a Boolean networkcan determine properties of the asymptotic dynamics. For instance, consideringthe asynchronous dynamics, the absence of positive cycles guarantees theexistence of a unique attractor, and the absence of negative cycles ensuresthat all attractors are fixed points. In presence of multiple attractors, onemight be interested in properties that ensure that attractors are sufficiently"isolated", that is, they can be found in separate subspaces or even trapspaces, subspaces that are closed with respect to the dynamics. Here weintroduce notions of separability for attractors and identify correspondingnecessary conditions on the interaction graph. In particular, we show that ifthe interaction graph has at most one positive cycle, or at most one negativecycle, or if no positive cycle intersects a negative cycle, then the attractorscan be separated by subspaces. If the interaction graph has no path from anegative to a positive cycle, then the attractors can be separated by trapspaces. Furthermore, we study networks with interaction graphs admitting twovertices that intersect all cycles, and show that if their attractors cannot beseparated by subspaces, then their interaction graph must contain a copy of thecomplete signed digraph on two vertices, deprived of a negative loop. We thusestablish a connection between a dynamical property and a complex networkmotif. The topic is far from exhausted and we conclude by stating some openquestions.

Attractor separation and signed cycles in asynchronous Boolean networks