A NEW CONSTRUCTION FOR VERTEX DECOMPOSABLE GRAPHS

Abstract

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 $Gsetminus S$ 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.