I have an arbitrary directed, non-complete graph. I would like to find a maximal subset of nodes within that graph such that the induced subgraph has no cycles. I'm sure there must be a name for this problem in the literature - do you know what it is?
This is the "maximum acyclic subgraph" problem (just search Google).
I once used such a thing for an info-theory bound on an "index coding" problem (but for a directed graph in that case), Lemma 1 here: