Maintenance of a minimum spanning forest in a dynamic planar graph

We give efficient algorithms for maintaining a minimum spanning forest of a planar graph subject to on-line modifications. The modifications supported include changes in the edge weights, and insertion and deletion of edges and vertices. To implement the algorithms, we develop a data structure called an edge-ordered dynamic tree, which is a variant of the dynamic tree data structure of Sleator and Tarjan. Using this data structure, our algorithms run in O(logn) time per operation and O(n) space. The algorithms can be used to maintain the connected components of a dynamic planar graph in O(logn) time per operation.