Where in the brain consciousness resides remains unclear. It has been suggested that the subnetworks supporting consciousness should be bidirectionally (recurrently) connected because both feed-forward and feedback processing are necessary for conscious experience. Accordingly, evaluating which subnetworks are bidirectionally connected and the strength of these connections would likely aid the identification of regions essential to consciousness. Here, we propose a method for hierarchically decomposing a network into cores with different strengths of bidirectional connection, as a means of revealing the structure of the complex brain network. We applied the method to a whole-brain mouse connectome. We found that cores with strong bidirectional connections consisted of regions presumably essential to consciousness (e.g., the isocortical and thalamic regions, and claustrum) and did not include regions presumably irrelevant to consciousness (e.g., cerebellum). Contrarily, we could not find such correspondence between cores and consciousness when we applied other simple methods which ignored bidirectionality. These findings suggest that our method provides a novel insight into the relation between bidirectional brain network structures and consciousness. Our recent preprint on this work is here: https://doi.org/10.1101/2021.07.12.452022.