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Recent advances in scale-free networks have claimed that their topologies are very weak against attacks. The inhomogeneous connectivity distribution of large-scale current communication networks, such as the Internet, could be exploited by evil hackers in order to damage these systems. However, there have not been many studies on the approaches and consequences of such targeted attacks. In this paper, we propose an in-depth study of the Internet topology robustness to attacks at the network layer. Several attacking techniques are presented, as well as their effects on the connectivity of the Internet. We show that although the removal of a small fraction of nodes (less than 10%) can damage the Internet connectivity, such a node removal attack would still require a large amount of work to be carried out. To achieve this, we study in detail the interactions between the intradomain and interdomain levels of the Internet through the use of an overlay.