Galaxy cutsets in graphs

13 years 3 months ago

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Given a network G = (V, E), we say that a subset of vertices S ⊆ V has radius r if it is spanned by a tree of depth at most r. We are interested in determining whether G has a cutset that can be written as the union of k sets of radius r. This generalizes the notion of k-vertex connectivity, since in the special case r = 0, a set spanned by a tree of depth at most r is a single vertex. Our motivation for considering this problem is that it constitutes a simple model for virus-like malicious attacks on G: An attack occurs at a subset of k vertices and begins to spread through the network. Any vertex within distance r of one of the initially attacked vertices may become infected. Thus an attack corresponds to a subset of vertices that is spanned by k trees of depth at most r. The question we focus on is whether a given network has a cutset of this particular form. The main results of this paper are the following. If r = 1, an attack corresponds to a subset of vertices which is the unio...

Nicolas Sonnerat, Adrian Vetta

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