A new construction for vertex decomposable graphs

Let G be a finite simple graph on the vertex set V(G) and let S subseteq V(G) . Adding a whisker to G at x means adding a new vertex y and edge xy to G where x in V(G) . The graph Gcup W(S) is obtained from G by adding a whisker to every vertex of S. We prove that if G setminus S is either a graph with no chordless cycle of length other than 3 or 5, chordal graph or C_5, then G cup W(S) is a vertex decomposable graph.