A characterization for 2-self-centered graphs

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 compliments. 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 intersecting bi-independent covering (IBIC) and a structure named generalized complete bipartite graph (GCBG). Then, we complete characterization by characterizing edge-minimal 2-self-centered graphs with some triangles. Hence, the main charactrization is done since a graph is 2-selfcentered 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.