Some new central parts of connected graphs

TL;DR

This paper extends the concepts of characteristic set, subtree core, and core vertices from trees to general connected graphs, exploring their properties, differences, and relationships with other central parts.

Contribution

It introduces the generalized central parts for connected graphs, proves their possible differences, and characterizes their behavior in vertex transitive graphs and supergraphs.

Findings

In a connected graph, all six central parts can differ.

In vertex transitive graphs, all six central parts coincide as the entire vertex set.

Any graph can be embedded in a connected supergraph with the whole vertex set as the characteristic center.

Abstract

The center, median and the security center are three central parts defined for any connected graph whereas the characteristic set, subtree core and core vertices are three central parts defined for trees only. We extend the concept of the characteristic set, subtree core and core vertices to general connected graphs and call them the characteristic center, subgraph core and core vertices, respectively. We show by examples that in a connected graph all the above six central parts can be different and also prove that for a connected vertex transitive graph each of the six central parts is the whole vertex set. Further it is shown that given any graph $G$, there exists a connected supergraph $G_{ch}$ of $G$ with the whole vertex set of $G$ as the characteristic center. Associated with the subgraph core and core vertices, we leave some unanswered question related to the graph centrality.