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DISMANTLING SPARSE RANDOM GRAPHS
, 2007
"... We consider the number of vertices that must be removed from a graph G in order that the remaining subgraph has no component with more than k vertices. Our principal observation is that, if G is a sparse random graph or a random regular graph on n vertices with n → ∞, then the number in question is ..."
Abstract

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We consider the number of vertices that must be removed from a graph G in order that the remaining subgraph has no component with more than k vertices. Our principal observation is that, if G is a sparse random graph or a random regular graph on n vertices with n → ∞, then the number in question is essentially the same for all values of k that satisfy both k → ∞ and k = o(n). The process of removing vertices from a graph G so that the remaining subgraph has only small components is known as fragmentation. Typically, the aim is to remove the least possible number of vertices to achieve a given component size; this is equivalent to determining the largest induced subgraph whose components are at most that size. This process has been studied in (at least) two different lines of research, from different perspectives and with quite different component sizes. In this note we point out that, as far as sparse random graphs are concerned, these two perspectives actually arrive at the same answer.
The symmetry in the martingale inequality
, 2001
"... In this paper, we establish a martingale inequality and develop the symmetry argument to use this martingale inequality. We apply this to the length of the longest increasing subsequences and the independence number ..."
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In this paper, we establish a martingale inequality and develop the symmetry argument to use this martingale inequality. We apply this to the length of the longest increasing subsequences and the independence number