Improved Data Structures for Fully Dynamic Biconnectivity

We present fully dynamic algorithms for maintaining the biconnected components in general and plane graphs and lower bounds for fully dynamic k-edge connectivity, k-vertex connectivity, and planarity testing. Fully dynamic algorithms maintain a graph during a sequence of insertions and deletions of edges or isolated vertices. Let m be the number of edges and n be the number of vertices in a graph. The time per operation of the best known algorithms are 0(@) in general graphs and O(log n) in plane graphs for fully dynamic connectivity and O(min{m2/3, n}) in general graphs and O(@) in plane graphs for fully dynamic biconnectivity. We improve the later running times to O(min{JiZlog n, n}) in general graphs and 0(log2 n) in plane graphs. In general graphs the update time is amortized and our algom”thm can also find the biconnected components of all vertices in time O(n). We also prove lower bounds for the complexity of maintaining fully dynamic k-edge or k-vertex connectivity in plane and in (k-l) -vertex connected graphs for any constant k and for fully dynamic planarity testing. We show an amortized lower bound of C2(log n/log log n) per operation in the cell probe model. These are the jirst lower bounds for dynamic connectivity problems.