Controlling an epidemic without a vaccine or treatment requires the limitation of transmission contacts. Here we use MCMC to reduce the number of removed contacts required to halt the spread of a disease depending on the network topology and the number of links removed. Analyzing the resulting ensembles, we find that the high epidemic threshold correlates with a more persistently fully connected network, but smaller giant component when it finally disconnects, as well as a more sharply peaked degree distribution. A similar effect is also achieved by directly homogenizing the node degrees in the network. We show that using such a homogenization as a removal strategy efficiently limits spreading. The resulting removal strategy performs comparably in halting the epidemic as MCMC, while operating fully on local information available to every node.