Geodesic
A Characterization for 2-Self-Centered Graphs
Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 1, pp. 27-37.

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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.
Keywords: self-centered graphs, specialized bi-independent covering (SBIC), generalized complete bipartite graphs (GCB) 
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Shekarriz, Mohammad Hadi; Mirzavaziri, Madjid; Mirzavaziri, Kamyar. A Characterization for 2-Self-Centered Graphs. Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 1, pp. 27-37. http://geodesic.mathdoc.fr/item/DMGT_2018_38_1_a1/

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