The paper gives a connectivity oracle under k vertex failures with O(k^6) n-independent update time, O(k) query time, and near-linear space and preprocessing for constant k.
Journal of Algorithms , volume=
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Connectivity Oracle Under Vertex Failures by Shortcutting Unbreakable Decomposition
The paper gives a connectivity oracle under k vertex failures with O(k^6) n-independent update time, O(k) query time, and near-linear space and preprocessing for constant k.