Topology-hiding computation (THC) is a form of multi-party computation over an incomplete communication graph that maintains the privacy of the underlying graph topology. In a line of recent works [Moran, Orlov & Richelson TCC’15, Hirt et al. CRYPTO’16, Akavia & Moran EUROCRYPT’17, Akavia et al. CRYPTO’17], THC protocols for securely computing any function in the semi-honest setting have been constructed. In addition, it was shown by Moran et al. that in the fail-stop setting THC with negligible leakage on the topology is impossible. In this paper, we further explore the feasibility boundaries of THC. We show that even against semi-honest adversaries, topology-hiding broadcast on a small (4-node) graph implies oblivious transfer; in contrast, trivial broadcast protocols exist unconditionally if topology can be revealed.We strengthen the lower bound of Moran et al. identifying and extending a relation between the amount of leakage on the underlying graph topology that must be revealed in the fail-stop setting, as a function of the number of parties and communication round complexity: Any n-party protocol leaking bits for must have rounds. We then present THC protocols providing close-to-optimal leakage rates, for unrestricted graphs on n nodes against a fail-stop adversary controlling a dishonest majority of the n players. These constitute the first general fail-stop THC protocols. Specifically, for this setting we show: A THC protocol that leaks at most one bit and requires rounds.A THC protocol that leaks at most bits for arbitrarily small non-negligible, and requires rounds. These protocols also achieve full security (with no leakage) for the semi-honest setting. Our protocols are based on one-way functions and a (stateless) secure hardware box primitive. This provides a theoretical feasibility result, a heuristic solution in the plain model using general-purpose obfuscation candidates, and a potentially practical approach to THC via commodity hardware such as Intel SGX. Interestingly, even with such hardware, proving security requires sophisticated simulation techniques.