Preimage problems for deterministic finite automata

Given a subset of states $S$ deterministic finite automaton and word $w$, the preimage is all mapped to state in by action $w$. We study three natural problems concerning words giving certain preimages. The first problem whether, for given subset, there exists \emph{extending} (giving larger preimage). second whether \emph{totally extending} whole set as preimage)---equivalently, an \emph{avoiding} complementary subset. third \emph{resizing} word. also consider variants where length upper bounded, size restricted, strongly connected, synchronizing, or binary. conclude with summary complexities combinations cases.