We provide data strutures that maintain a graph as edges are inserted and deleted, and keep track of the following properties with the following times: minimum spanning forests, graph connectivity, graph 2-edge connectivity, and bipartiteness in time O ( n 1/2 ) per change; 3-edge connectivity, in time O ( n 2/3 ) per change; 4-edge connectivity, in time O ( n α( n )) per change; k -edge connectivity for constant k , in time O ( n log n ) per change;2-vertex connectivity, and 3-vertex connectivity, in the O ( n ) per change; and 4-vertex connectivity, in time O ( n α( n )) per change. Further results speed up the insertion times to match the bounds of known partially dynamic algorithms. All our algorithms are based on a new technique that transforms an algorithm for sparse graphs into one that will work on any graph, which we call sparsification.

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We study novel combinatorial properties of graphs that allow us to devise a completely new approach to dynamic all pairs shortest paths problems. Our approach yields a fully dynamic algorithm for general directed graphs with non-negative real-valued edge weights that supports any sequence of operations in Õ(n2)1 amortized time per update and unit worst-case time per distance query, where n is the number of vertices. We can also report shortest paths in optimal worst-case time. These bounds improve substantially over previous results and solve a long-standing open problem. Our algorithm is deterministic and uses simple data structures

We present the first deterministic data structures for maintaining approximate minimum vertex cover and maximum matching in a fully dynamic graph G = (V, E), with |V | = n and |E| = m, in o( √ m ) time per update. In particular, for minimum vertex cover we provide deterministic data structures for maintaining a (2+ ) approximation in O(log n/ 2 ) amortized time per update. For maximum matching, we show how to maintain a (3 + ) approximation in O(min( √ n/ , m 1/3 / 2 )) amortized time per update, and a (4 + ) approximation in O(m 1/3 / 2 ) worst-case time per update. Our data structure for fully dynamic minimum vertex cover is essentially near-optimal and settles an open problem by Onak and Rubinfeld [13]. * An extended abstract of this paper, not containing the algorithm in Section 4 and not containing the proof of Theorem 2.11, has been accepted for publication in ACM-SIAM Symposium on Discrete Algorithms (SODA)' 2015.

Edge and vertex connectivity are fundamental concepts in graph theory. While they have been thoroughly studied in the case of undirected graphs, surprisingly not much has been investigated for directed graphs. In this paper we study 2-edge connectivity problems in directed graphs and, in particular, we consider the computation of the following natural relation: We say that two vertices v and w are 2-edge-connected if there are two edge-disjoint paths from v to w and two edgedisjoint paths from w to v. This relation partitions the vertices into blocks such that all vertices in the same block are 2-edge-connected. Differently from the undirected case, those blocks do not correspond to the 2-edge-connected components of the graph. The main result of this paper is an algorithm for computing the 2-edge-connected blocks of a directed graph in linear time. Besides being asymptotically optimal, our algorithm improves significantly over previous bounds. Once the 2-edge-connected blocks are available, we can test in constant time if two vertices are 2-edge-connected. Additionally, we also show how to compute in linear time a sparse certificate for this relation, i.e., a subgraph of the input graph that has O(n) edges and maintains the same 2-edge-connected blocks as the input graph, where n is the number of vertices.