K. Mirzavaziri, M. Mirzavaziri, M.H. Shekarriz
A characterization for 2-self-centered graphs
Discussiones Mathematicae Graph Theory
Received 30.11.2015, Revised 24.08.2016, Accepted 01.09.2016, doi: 10.7151/dmgt.1994

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 specialized bi-independent covering (SBIC) 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 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.
self-centered graphs, specialized bi-independent covering (SBIC), generalized complete bipartite graphs (GCB)