Fully dynamic maintainance (both inserts and deletes) of bridge blocks and blocks in amortized polylog time using Top trees. Fully dynamic maintainance of components in $O(\log^2 n)$ amortized time.
Global search on Top trees and its application on maintaing tree center in $O(\log n)$ time.
Simillar problems for plan(e/ar) and/or directed graphs. Depends on current state of art dealing with dynamic graph problems.
Report seminar concerning problems from TIN023.