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.
|Title of host publication||Automata, Languages and Programming - 37th International Colloquium, ICALP 2010, Proceedings|
|Number of pages||12|
|State||Published - 2010|
|Event||37th International Colloquium on Automata, Languages and Programming, ICALP 2010 - Bordeaux, France|
Duration: 6 Jul 2010 → 10 Jul 2010
|Name||Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)|
|Conference||37th International Colloquium on Automata, Languages and Programming, ICALP 2010|
|Period||6/07/10 → 10/07/10|
Bibliographical noteFunding Information:
E-mail address: firstname.lastname@example.org (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.
- Tutte polynomial
- network reliability
- sparse subgraphs