The integrity of a cubic graph

Integrity, a measure of network reliability, is de-ned as I(G) = min S⊂V {|S|+ m(G − S)}; where G is a graph with vertex set V and m(G− S) denotes the order of the largest component of G − S. We prove an upper bound of the following form on the integrity of any cubic graph with n vertices: I(G)¡ 3 n+ O (√ n ) : Moreover, there exist an in-nite family of connected cubic graphs whose integrity satis-es a linear lower bound I(G)¿ n for some constant . We provide a value for , but it is likely not best possible. To prove the upper bound we -rst solve the following extremal problem. What is the least number of vertices in a cubic graph whose removal results in an acyclic graph? The solution (with a few minor exceptions) is that n=3 vertices su5ce and this is best possible. ? 2003 Elsevier B.V. All rights reserved.