### Abstract

A communication network can be modeled as a graph where the nodes of the graph represent the sites, and the edges represent the links between the sites. The edges of the graph operate with equal probability p. The all-terminal reliability of the network with n nodes and b edges can be written as where F_{i}is the number of connected subgraphs with (b - i) edges and q= (1 -p). All known subgraph counting bounds for R use exact values for F_{0}, …, F_{c}and F_{d}where c is the cardinality of a minimum cut and d=b - n + 1. In this paper we derive upper and lower bounds for R for planar networks by using exact values of F_{d-1}, F_{d-2}, F_{d-3}, F_{c+1}and F_{c+2}where approximations were used before. The effect of using these exact values instead of approximations on Kruskal-Katona bounds and Ball-Provan bounds is studied.

Original language | English (US) |
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Pages (from-to) | 261-273 |

Number of pages | 13 |

Journal | North-Holland Mathematics Studies |

Volume | 144 |

Issue number | C |

DOIs | |

State | Published - Jan 1987 |

Externally published | Yes |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*North-Holland Mathematics Studies*,

*144*(C), 261-273. https://doi.org/10.1016/S0304-0208(08)73060-3