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A significant challenge for researchers analysing the Internet AS-level topology graph is how to interpret the global organization of the graph as the coexistence of its structural blocks (communities) associated with more highly interconnected parts. While a huge number of papers have already been published on the issue of community detection, very little attention has so far been devoted to the discovery and interpretation of Internet communities at the various levels of abstractions (e.g. Autonomous System level, Point of Presence level). We believe that by discovering and interpreting a priori these unknown building blocks (i.e. communities), this will then pave the way for new types of analysis which are crucial in understanding of the structural and functional properties of the Internet at least at the AS level of abstraction. We thus propose a novel type of analysis of the Internet AS-level topology graph by exploiting the k-clique community definition. First, we show that detected communities can be described by a tree representation. Then we show the presence of two classes of k-clique communities: those that are strictly affected by the nesting process which is embedded in the k-clique community definition, and, on the other hand, those that appear as branches in the tree. We conclude our analysis by highlighting the properties that characterize k-clique communities with different k values by exploiting both geographical data and information related to Internet exchange Points (IXPs).