Separator based sparsification for dynamic planar graph algorithms

We describe algorithms and data structures for maintaining a dynamic planar graph subject to edge insertions and edge deletions that preserve planarity but that can change the embedding. We give a fully dynamic planarity testing algorithm that maintains a graph subject to edge insertions and deletions, and allows queries that test whether the graph is currently planar, or whether a potential new edge would violate planarity, in amortized time O(nl 12) per update or query. We maintain the 2and 3-vertex-connected components, and the 3and 4-edge-connected components of a planar graph in O(n.llz ) time per insertion, deletion or query. We give fully dynamic algorithms for maintaining the connected components, the 2-edge-connected components, and the minimum spanning forest of a planar graph in time (9(log n) per insertion and 0(log2 n) per deletion, assuming that insertions keep the graph planar. All our algorithms improve previous bounds: the improvements are based upon a new type of sparsification combined wit h several properties of separators in planar graphs.