Improved Data Structures for Fully Dynamic Biconnectivity

Abstract

We present fully dynamic algorithms for maintaining the biconnected components in general and plane graphs. A fully dynamic algorithm maintains 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 deterministic algorithms is $O(\sqrt n)$ in general graphs and O(log n) in plane graphs for fully dynamic connectivity and O(min m2/3 ,n}) in general graphs and $O(\sqrt n)$ in plane graphs for fully dynamic biconnectivity. We improve the later running times to $O(\sqrt {m\log n})$ in general graphs and O(log 2n ) in plane graphs. Our algorithm for general graphscan also find the biconnected components of all vertices in time O(n).