For a set S of vertices and the vertex v in a connected graph G, max
x2S
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, Km,n, Kn −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