TY - JOUR

T1 - Sparse reliable graph backbones

AU - Chechik, Shiri

AU - Emek, Yuval

AU - Patt-Shamir, Boaz

AU - Peleg, David

N1 - Funding Information:
E-mail address: boaz@eng.tau.ac.il (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.

PY - 2012/1

Y1 - 2012/1

N2 - 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.

AB - 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.

KW - Network reliability

KW - Sparse subgraphs

KW - Tutte polynomial

UR - http://www.scopus.com/inward/record.url?scp=82355187656&partnerID=8YFLogxK

U2 - 10.1016/j.ic.2011.10.007

DO - 10.1016/j.ic.2011.10.007

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AN - SCOPUS:82355187656

SN - 0890-5401

VL - 210

SP - 31

EP - 39

JO - Information and Computation

JF - Information and Computation

ER -