An O(n^{3/2})-size subgraph preserves 2-approximate min-cost arborescences under single edge faults with fast updates, plus a tight k times rank bound for k-fault-tolerant matroid preservers.
Fault-tolerant subgraph for single-source reachability: General and optimal.SIAM Journal on Computing, 47(1):80–95, 2018
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Low-Cost Arborescence Under Edge Faults
An O(n^{3/2})-size subgraph preserves 2-approximate min-cost arborescences under single edge faults with fast updates, plus a tight k times rank bound for k-fault-tolerant matroid preservers.