A Graph is called 2-self-centered if its diameter and radius both equal to 2.
In this paper, we begin characterizing these graphs by characterizing
edge-maximal 2-self-centered graphs via their complements. Then we split
characterizing edge-minimal 2-self-centered graphs into two cases. First, we
characterize edge-minimal 2-self-centered graphs without triangles by
introducing \emph{specialized bi-independent covering (SBIC)} and a structure
named \emph{generalized complete bipartite graph (GCBG)}. Then, we complete
characterization by characterizing edge-minimal 2-self-centered graphs with
some triangles. Hence, the main characterization is done since a graph is
2-self-centered if and only if it is a spanning subgraph of some edge-maximal
2-self-centered graphs and, at the same time, it is a spanning supergraph of
some edge-minimal 2-self-centered graphs