Relationships between vertex attack tolerance and other vulnerability parameters

Abstract

Let G ( V,E ) be a simple undirected graph. Recently, the vertex attack tolerance (VAT) of G has been defined as τ( G ) = min {| S | / | V - S - C max ( G - S )|+1 : S ⊂ V } , where C max ( G − S ) is the order of a largest connected component in G − S . This parameter has been used to measure the vulnerability of networks. The vertex attack tolerance is the only measure that fully captures both the major bottlenecks of a network and the resulting component size distribution upon targeted node attacks. In this article, the relationships between the vertex attack tolerance and some other vulnerability parameters, namely connectivity, toughness, integrity, scattering number, tenacity, binding number and rupture degree have been determined.