A Characterization For 2-Self-Centered Graphs

TL;DR

This paper characterizes 2-self-centered graphs, which have diameter and radius both equal to 2, by analyzing their maximal and minimal structures, including special cases with and without triangles.

Contribution

It provides the first comprehensive characterization of 2-self-centered graphs, including edge-maximal and edge-minimal cases, using new concepts like SBIC and GCBG.

Findings

Characterization of edge-maximal 2-self-centered graphs via complements.

Introduction of SBIC and GCBG for classifying triangle-free 2-self-centered graphs.

Complete characterization of all 2-self-centered graphs through spanning subgraphs and supergraphs.

Abstract

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.