Biconnectivity, st-numbering and other applications of DFS using O(n) bits.

We consider space efficient implementations of some classical applications of DFS including the problem of testing biconnectivity and 2-edge connectivity, finding cut vertices and cut edges, computing chain decomposition and st-numbering of a given undirected graph G on n vertices and m edges. Classical algorithms for them typically use DFS and some Ω(lgn) bits1of information at each vertex. Building on a recent O(n)-bits implementation of DFS due to Elmasry et al. (STACS 2015) we provide O(n)-bit implementations for all these applications of DFS. Our algorithms take O(mlgcnlglgn) time for some small constant c (where c≤2). Central to our implementation is a succinct representation of the DFS tree and a space efficient partitioning of the DFS tree into connected subtrees, which maybe of independent interest for designing other space efficient graph algorithms.