If I reverse a single edge in a DAG, how can I efficiently propagate the effects of that (e.g. change other edges) such that I don't introduce any cycles by said reversal? Is there a named algorithm for this that I should look up? I see some other questions about reversing an entire DAG; I'm only concerned with a single edge plus its necessary influence, which I hope is not the entire graph in a typical case.
To add more context: I'm working with some disjunctive job-shop representations. Example: https://acrogenesis.com/or-tools/documentation/user_manual/manual/ls/jobshop_def_data.html . I want to minimize the number of updated edges. And after the update, I still need all of the nodes in the graph to be accessible through some path.