 Edge deletion preserves recurrence

Deleting an edge of a locally finite recurrent graph leaves recurrent Markov chain connected components. On an infinite component choose a root away from the deleted edge's endpoints, so its vertex degree does not change. The Rayleigh monotonicity principle increases resistance to wired outer boundaries, and effective resistance to infinity therefore remains infinite. Finite components are recurrent Markov chain too. This is a recurrence-type comparison; at an endpoint, changes in the vertex degree factor prevent a blanket monotonicity claim about first-return probabilities.