Sparse reliable graph backbones

Shiri Chechik, Yuval Emek, Boaz Patt-Shamir, David Peleg

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

2 Scopus citations


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.

Original languageEnglish
Title of host publicationAutomata, Languages and Programming - 37th International Colloquium, ICALP 2010, Proceedings
Number of pages12
EditionPART 2
StatePublished - 2010
Externally publishedYes
Event37th International Colloquium on Automata, Languages and Programming, ICALP 2010 - Bordeaux, France
Duration: 6 Jul 201010 Jul 2010

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
NumberPART 2
Volume6199 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference37th International Colloquium on Automata, Languages and Programming, ICALP 2010

Bibliographical note

Funding Information:
E-mail address: (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


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