Abstract
Integrity, a measure of network reliability, is defined 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)< 1 3 n+ O n . Moreover, there exist an infinite family of connected cubic graphs whose integrity satisfies a linear lower bound I( G)> βn for some constant β. We provide a value for β, but it is likely not best possible. To prove the upper bound we first 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 suffice and this is best possible.
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