  Mutually connected components (MCCs) play an important role as a measure of
resilience in the study of interdependent networks. Despite their importance,
an efficient algorithm to obtain the statistics of all MCCs during the removal
of links has thus far been absent. Here, using a well-known fully dynamic graph
algorithm, we propose an efficient algorithm to accomplish this task. We show
that the time complexity of this algorithm is approximately $O({N^{1.2} })$ for
random graphs, which is more efficient than $O(N^{2})$ of the brute-force
algorithm. We confirm the correctness of our algorithm by comparing the
behavior of the order parameter as links are removed with existing results for
three types of double-layer multiplex networks. We anticipate that this
algorithm will be used for simulations of large-size systems that have been
previously inaccessible.
