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.
@article{bwmeta1.element.doi-10_7151_dmgt_1994,
author = {Mohammad Hadi Shekarriz and Madjid Mirzavaziri and Kamyar Mirzavaziri},
title = {A Characterization for 2-Self-Centered Graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {38},
year = {2018},
pages = {27-37},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1994}
}
Mohammad Hadi Shekarriz; Madjid Mirzavaziri; Kamyar Mirzavaziri. A Characterization for 2-Self-Centered Graphs. Discussiones Mathematicae Graph Theory, Tome 38 (2018) pp. 27-37. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1994/