Deterministic pushdown automata as well as deterministic one-counter automata with the restriction that their underlying state graph is quasi-acyclic are investigated. Here, quasi-acyclicity means that the graph is allowed to have self-loops, but removing the self-loops leads to an acyclic graph. Such automata are called state-freezing, since a state left can never be entered again. It turns out that the state-freezing property is a strong property, since the considered automata with the state-freezing property are less powerful than in the general case. This result holds in the real-time case, i.e., \(\lambda \) -moves are not allowed, as well as in the case of allowed \(\lambda \) -moves. We study the closure properties of the corresponding language families and can show that all families are anti-AFLs, whereas all families are closed under complementation. Finally, we look at decidability questions and obtain the undecidability of the inclusion problem already in the case of state-freezing real-time deterministic pushdown automata. However, the status of the inclusion problem in the case of state-freezing deterministic one-counter automata remains open.