I have a problem that is reducible to (efficiently) determining the reachability of a node in a fully dynamic planar digraph.

I'm aware of ["A fully dynamic data structure for reachability in planar digraphs"][1] which provides $O(n^{2/3} \log n)$ query with a $O(n)$-space data structure.

Can this be / has this been bettered?

If all my queries have the same source node, is there a more efficient (in time/space/both) way?

Are there any other related literature that deals with more efficient queries albeit with more restrictions imposed on the digraph?

Thanks!


  [1]: https://doi.org/10.1007/3-540-57273-2_72 "Subramanian, S. (1993). In: Lengauer, T. (eds) Algorithms—ESA '93. ESA 1993. Lecture Notes in Computer Science, vol 726. Springer, Berlin, Heidelberg. zbMATH review of symposium proceedings at https://zbmath.org/?q=an:0825.00137"