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SIAM Journal on Computing
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STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
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Given a connected graph G and a failure probability p(e) for each edge e in G, the reliability of G is the probability that G remains connected when each edge e is removed independently with probability p(e). In this paper it is shown that every n-vertex graph contains a sparse backbone, i.e., a spanning subgraph with O(n log n) edges whose reliability is at least (1-n-Ω(1)) times that of G. Moreover, for any pair of vertices s, t in G, the (s, t)-reliability of the backbone, namely, the probability that s and t remain connected, is also at least (1-n-Ω(1)) times that of G. Our proof is based on a polynomial time randomized algorithm for constructing the backbone. In addition, it is shown that the constructed backbone has nearly the same Tutte polynomial as the original graph (in the quarterplane x ≥ 1, y 1), and hence the graph and its backbone share many additional features encoded by the Tutte polynomial.