Testing connectivity of faulty networks in sublinear time

Given a set F of vertices of a connected graph G, we study the problem of testing the connectivity of G−F in polynomial time with respect to |F | and the maximum degree ∆ of G. We present two approaches. The first algorithm for this problem runs in O ( |F |∆2ε−1 log(|F |∆ε−1) ) time for every graph G with vertex expansion at least ε > 0. The other solution, designed for the class of graphs with cycle basis consisting of cycles of length at most l, leads to O ( |F |∆dl/2e log(|F |∆bl/2c) ) running time. We also present an extension of this method to test the biconnectivity of G− F in O ( |F |∆ log(|F |∆) ) time.