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(nlogn) 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 quarter-plane x≥1, y>1), and hence the graph and its backbone share many additional features encoded by the Tutte polynomial.
Bibliographical noteFunding Information:
E-mail address: email@example.com (B. Patt-Shamir). 1 Supported in part by Israel Science Foundation (grant 1372/09) and by the Israel Ministry of Science and Technology. 2 Supported in part by the Israel Ministry of Science and Technology.
- Network reliability
- Sparse subgraphs
- Tutte polynomial