  The study of the graph diameter of polytopes is a classical open problem in
polyhedral geometry and the theory of linear optimization. In this paper we
continue the investigation initiated in [4] by introducing a vast hierarchy of
generalizations to the notion of graph diameter. This hierarchy provides some
interesting lower bounds for the usual graph diameter. After explaining the
structure of the hierarchy and discussing these bounds, we focus on clearly
explaining the differences and similarities among the many diameter notions of
our hierarchy. Finally, we fully characterize the hierarchy in dimension two.
It collapses into fewer categories, for which we exhibit the ranges of values
that can be realized as diameters.
