Bounds for All-Terminal Reliability in Planar Networks

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_iis the number of connected subgraphs with (b - i) edges and q= (1 -p). All known subgraph counting bounds for R use exact values for F₀, …, F_cand F_dwhere 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+1and F_c+2where approximations were used before. The effect of using these exact values instead of approximations on Kruskal-Katona bounds and Ball-Provan bounds is studied.