Discrete event systems are increasingly used as a modeling tool to assess safety and cybersecurity of complex systems. In both cases, the analysis relies on the extraction of critical sequences. This approach proves to be very powerful. It suffers, however, from the combinatorial explosion of the number of sequences to look at. To push the limits of what is feasible with reasonable computational resources, extraction algorithms use cutoffs and minimality criteria. In this article, we review the principles of extraction algorithms, and we show that there are important differences between critical sequences extracted in the context of safety analyses and those extracted in the context of cybersecurity analyses. Based on this thorough comparison, we introduce a new cutoff criterion, so-called footprint, that aims at capturing the willfulness of an intruder performing a cyberattack. We illustrate our presentation by means of three case studies, one focused on the analysis of failures and two focused on the analysis of cyberattacks and their effects on safety. We show experimentally the interest of the footprint criterion.