The eccentricity e(u) of a vertex u is the maximum distance of u to any other vertex of G. The maximum and the minimum eccentricity among the vertices of a graph G are known as the diameter and the radius of G respectively. If they are equal then the graph is said to be a self - centered graph. Edge addition /extension to a graph either retains or changes the parameter of a graph, under consideration. In this paper mainly, we consider edge extension for cycles, with respect to the self-centeredness(of cycles),that is, after an edge set is added to a self centered graph the resultant graph is also a self-centered graph. Also, we have other structural results for graphs with edge -extensions.