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 logn) 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. © 2010 Springer-Verlag Berlin Heidelberg.
CITATION STYLE
Chechik, S., Emek, Y., Patt-Shamir, B., & Peleg, D. (2010). Sparse reliable graph backbones. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 6199 LNCS, pp. 261–272). https://doi.org/10.1007/978-3-642-14162-1_22
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