Structural Oscillatority Criterion of Boolean Networks

Azuma et al. showed that a cactus-expandable Boolean network $\mathrm{\Sigma }$ is structurally oscillatory if and only if all the simple cycles of $\mathrm{\Sigma }$ contain an odd number of inhibiting edges. We show that we do not need the cactus-expandable condition. Namely, a strongly connected Boolean network $\mathrm{\Sigma }$ is structurally oscillatory if and only if all the simple cycles of $\mathrm{\Sigma }$ contain an odd number of inhibiting edges. Additionally, we provide a characterization of a structurally oscillatory Boolean network for the general case.