Given a non empty set S of vertices of a graph, the partiality of a vertex with respect to S
is the di erence between maximum and minimum of the distances of the vertex to the vertices
of S. The vertices with minimum partiality constitute the fair center of the set. Any vertex set
which is the fair center of some set of vertices is called a fair set. In this paper we prove that the
induced subgraph of any fair set is connected in the case of trees and characterise block graphs
as the class of chordal graphs for which the induced subgraph of all fair sets are connected. The
fair sets of Kn, Km;n, Kne, wheel graphs, odd cycles and symmetric even graphs are identi ed.
The fair sets of the Cartesian product graphs are also discussed
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