On The Center Sets and Center Numbers of Some Graph Classes

TL;DR

This paper investigates the structure of center sets and introduces the concept of center number across various graph classes, providing enumeration and characterization results.

Contribution

It characterizes and enumerates center sets and center numbers for several graph classes, expanding understanding of graph center concepts.

Findings

Identified center sets for block graphs, complete bipartite graphs, wheel graphs, odd cycles, and symmetric even graphs.

Determined the center number for certain graph classes.

Provided enumeration methods for center sets in various graphs.

Abstract

For a set $S$ of vertices and the vertex $v$ in a connected graph $G$, $\displaystyle\max_{x \in S}d(x,v)$ is called the $S$-eccentricity of $v$ in $G$. The set of vertices with minimum $S$-eccentricity is called the $S$-center of $G$. Any set $A$ of vertices of $G$ such that $A$ is an $S$-center for some set $S$ of vertices of $G$ is called a center set. We identify the center sets of certain classes of graphs namely, Block graphs, $K_{m,n}$, $K_n-e$, wheel graphs, odd cycles and symmetric even graphs and enumerate them for many of these graph classes. We also introduce the concept of center number which is defined as the number of distinct center sets of a graph and determine the center number of some graph classes.