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 G\cup 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 ۳ or ۵, chordal graph or C_۵, then G \cup W(S) is a vertex decomposable graph.

Keywords:

vertex decomposable , shellabel , Cohen-Macaulay