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This incremental approach features delta preservation as it allows to restrict the generation of graph repairs to delta-preserving graph repairs, which do not revert the additions and deletions of the most recent consistency-violating graph update. We specify consistency of graphs using the logic of nested graph conditions, which is equivalent to first-order logic on graphs. Technically, the incremental approach encodes if and how the graph under repair satisfies a graph condition using the novel data structure of satisfaction trees, which are adapted incrementally according to the graph updates applied. In addition to the incremental approach, we also present two state-based graph repair algorithms, which restore consistency of a graph independent of the most recent graph update and which generate additional graph repairs using a global perspective on the graph under repair. 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