Network optimization strategies for the process of synchronization have generally focused on the re-wiring or re-weighting of links in order to: (1) expand the range of coupling strengths that achieve synchronization, (2) expand the basin of attraction for the synchronization manifold, or (3) lower the average time to synchronization. A new optimization goal is proposed in seeking the minimum subset of the edge set of the original network that enables the same essential ability to synchronize. We call this type of minimal spanning subgraph an Essential Synchronization Backbone (ESB) of the original system, and we present two algorithms for computing this subgraph. One is by an exhaustive search and the other is a method of approximation for this combinatorial problem. The solution spaces that result from different choices of dynamical systems and coupling vary with the level of hierarchical structure present and also the number of interwoven central cycles. These may provide insight into synchronization as a process of sharing and transferring information. Applications can include the important problem in civil engineering of power grid hardening, where new link creation may be costly, but instead, the defense of certain key links to the functional process may be prioritized.
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