In this paper we study the node failure identification problem in undirected graphs by means of Boolean Network Tomography. We argue that vertex connectivity plays a central role. We show tight bounds on the maximal identifiability in a particular class of graphs, the Line of Sight networks. We prove slightly weaker bounds on arbitrary networks. Finally we initiate the study of maximal identifiability in random networks. We focus on two models: the classical Erdoes-Renyi model, and that of Random Regular graphs. The framework proposed in the paper allows a probabilistic analysis of the identifiability in random networks giving a tradeoff between the number of monitors to place and the maximal identifiability.
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