Let S be a dominating set of a graph G and RS^c Í V(G). The set RS^c is called a center-smooth 2-RS^c set of a center smooth graph G if |N(v)∩RS^c| ≥ 2 for every vertex vÎS. The center-smooth 2-RS^c number γ_2cs(G) of a graph G is the number of vertices in a center-smooth 2-RS^c set of G. In this paper, we introduce the new concept center-smooth 2-RScnumber. The center-smooth 2-RScnumber γ_2cs(G) of G is the number of vertices in a center-smooth 2-RS^cset of G. Some results on this new parameter are established and γ_2cs(G) is computed for some special graphs and also proved that γ_2cs(G) = 6 for Petersen graph G. A result is proved for a triangle free connected graph G with minimum degree δ(G) ≥ 2. The following results are also proved. (i). If a connected graph G has exactly one vertex of degree p -1, then γ_2cs(G) = γ_2cs( ) + ∆(G) and (ii). Let G be a graph with cut edge e = uv where u and v are only central vertices, (G) = 1. If γ_2cs(G) = p - |{u, v}|,  then ????(G) + γ_2cs(G) = p.

Keywords- Center smooth graph, Restrict S^c-set, Center smooth 1^c domination number, center smooth 2-RS^c number.

Mathematics Subject Classification 2010: 05C12.