On maximal 3-edge-connected subgraphs of undirected graphs
share
We show how to find and efficiently maintain maximal 3-edge-connected subgraphs in undirected graphs. In particular, we provide the following results. (1) Two algorithms for the incremental maintenance of the maximal 3-edge-connected subgraphs. These algorithms allow for vertex and edge insertions, interspersed with queries asking whether two vertices belong to the same maximal 3-edge-connected subgraph, and there is a trade-off between their time- and space-complexity. Specifically, the first algorithm has O(mα(m,n) + n^2log^2 n) total running time and uses O(n) space, where m is the number of edge insertions and queries, and n is the total number of vertices inserted starting from an empty graph. The second algorithm performs the same operations in faster O(mα(m,n) + n^2α(n,n)) time in total, using O(n^2) space. (2) A fully dynamic algorithm for maintaining information about the maximal k-edge-connected subgraphs for fixed k. Our update bounds are O(n√(n) log n) worst-case time for k>4 and O(n√(n) ) worst-case time for k∈{3,4}. In both cases, we achieve constant time for maximal k-edge-connected subgraph queries. (3) A deterministic algorithm for computing the maximal k-edge-connected subgraphs, for any fixed k>2, in O(m+n√(n) ) time. This result improves substantially on the previously best known deterministic bounds and matches (modulo log factors) the expected time of the best randomized algorithm for the same problem. (4) A linear-time algorithm for computing a spanning subgraph with O(nlog n) edges that has the same maximal k-edge-connected subgraphs as the original graph.
READ FULL TEXT