I'm interested in efficiently maintaining the set of strongly connected components (SCC) in a directed (unweighted) graph under edge deletions only. While searching for ways I came across an article [1] that uses a generalised version of EvenShiloach-trees (ES-trees) [2] to maintain shortest path trees for the graph.

I have looked into ES-trees and understand how they work in the setting of undirected graphs, however a modification should allows to maintain a shortest path tree in directed graphs [3] but I can't work out how to do it.

Does anyone know of an example how ES-trees can be modified to maintain a shortest path tree under edge deletion?

[1] Bernstein, A., Probst, M., & Wulff-Nilsen, C. (2019). Decremental strongly-connected components and single-source reachability in near-linear time.

[2] Even, S., & Shiloach, Y. (1979). An on-line edge-deletion problem (No. CS Technion report CS0154). Computer Science Department, Technion.

[3] Henzinger, M. R., & King, V. (1995, October). Fully dynamic biconnectivity and transitive closure. In Proceedings of IEEE 36th Annual Foundations of Computer Science (pp. 664-672). IEEE.


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