  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.
