Connectivity oracles for graphs subject to vertex failures

Abstract

We introduce new data structures for answering connectivity queries in graphs subject to batched vertex failures. Our deterministic structure processes a batch of d ≤ d failed vertices in O (d3) time and thereafter answers connectivity queries in O(d) time. It occupies space O(dmlog n). We develop a randomized Monte Carlo version of our data structure with update time O (d2), query time O(d), and space O (m) for any d. This is the first connectivity oracle for general graphs that can efficiently deal with an unbounded number of vertex failures. Our data structures are based on a new decomposition theorem for an undirected graph G = (V;E), which is of independent interest. It states that for any terminal set U V we can remove a set B of U =(s - 2) vertices such that the remaining graph contains a Steiner forest for U - B with maximum degree s.