## Abstract

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 language | English |
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Title of host publication | Automata, Languages and Programming - 37th International Colloquium, ICALP 2010, Proceedings |

Pages | 261-272 |

Number of pages | 12 |

Edition | PART 2 |

DOIs | |

State | Published - 2010 |

Externally published | Yes |

Event | 37th International Colloquium on Automata, Languages and Programming, ICALP 2010 - Bordeaux, France Duration: 6 Jul 2010 → 10 Jul 2010 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Number | PART 2 |

Volume | 6199 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 37th International Colloquium on Automata, Languages and Programming, ICALP 2010 |
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Country/Territory | France |

City | Bordeaux |

Period | 6/07/10 → 10/07/10 |

### Bibliographical note

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.

## Keywords

- Tutte polynomial
- network reliability
- sparse subgraphs