The Shields-Harary indices of vulnerability of a graph

In 1972 the late Allen Shields posed a striking conjecture, eventually confirmed by Schanuel [1], about finite sequences of positive numbers. Shields and the first author saw that the conjectured result gave the answer for paths to a question that could be asked about any graph. That realization has now germinated and produced a family of graph parameters that measure certain kinds of fortibility of graphs, viewed as networks of storage fortresses. We discuss that family of parameters.