Exterior vertices in graphs and digraphs

This is a survey of various results concerning the extremity of graphs and digraphs. Given a graph, there are various subsets of the vertex set that can be interpreted as the "extremity of the graph"; we investigate several of these: the periphery, the set of vertices eccentric to the center, and the margin. We ·generalize the concept of periphery and margin to digraphs. Topics covered include the following: finding families of graphs such that the vertices in the periphery are the same as the vertices that are eccentric to the center, finding conditions on the product of two graphs which insure that the resulting periphery and set of vertices eccentric to the center are equal, studying graphs where each vertex of the graph has exactly one eccentric vertex, and finding conditions which allow us to construct supergraphs of a given graph such that the periphery or margin of the supergraph is isomorphic to the given graph. This last topic is also investigated in the context of digraphs.