On defining number of subdivided certain graph

In a given graph G = (V,E), a set of vertices S with an assignment of colors to them is said to be a defining set of the vertex coloring of G if there exists a unique extension of the colors of S to a c ≥ x(G) coloring of the vertices of G. A defining set with minimum cardinality is called a minimum defining set and its cardinality is the defining number. Let G = [C.sub.n+1] be circulant graph with V (G) = {[v.sub.i], [v.sub.2], ... ,[v.sub.n+1]}. Let G' and G" be graphs obtained from G by subdividing of edges [v.sub.i][v.sub.i+1] 1 ≤ i ≤ n + 1 (mod n + 1) and all of edges of G respectively. In this note, we study the chromatic and the defining numbers of G, G' and G".