Dismantling Sparse Random Graphs

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. Let Γ be a class of graphs. The classes we shall mostly be interested in are the classes Ck, the class of graphs whose components have at most k vertices, and F , the class of forests. Given such a class Γ, we define N(G,Γ) := max{|S| : G[S] ∈ Γ}, where S is a subset of the vertices of G and G[S] denotes the subgraph of G induced by S. We also define ν(G,Γ) := N(G,Γ)/|G|, so that 0 ≤ ν(G,Γ) ≤ 1. (To make this always defined, we set N(G,Γ) = 0 if no induced subgraph of G belongs to Γ; equivalently, we may regard the empty graph with no vertices as an element of Γ.) Thus, for example, the size of a largest independent set in G is N(G, C1) = ν(G, C1)|G|. (This is known as the independence number.) Similarly, n−N(G,F) is the decycling number, see e.g. Karp [16]. In this notation, the study of fragmentation is the study of the parameter ν(G, Ck) for various values of k. From the point of view of graph theory, it is natural to consider ν(G, Ck) for some large but finite value of k, for graphs G in which the number of vertices n = |G| grows large. This study was initiated by Edwards and Farr [5; 7]. On the other hand, in the study Date: September 12, 2007. 2000 Mathematics Subject Classification. 05C80; 05C40, 92D30.