Algorithms for Dynamic Reeb Graphs
  We present an algorithm for updating the Reeb graph under fully dynamic changes of the function values. The basic event is the interchange of two consecutive vertex values. The algorithm updates the Reeb graph in $O(l g{n})$ worst-case deterministic time for each such interchange, where $l$ is an upper bound on the size of the star of the involved vertices, and g(n) is a worst-case bound for the dynamic graph connectivity problem. Moreover, we argue that $O(l)$ is a lower bound for this operation in general. 
Salman Parsa
cs.CG cs.DS