# A new construction for vertex decomposable graphs

Document Type: Research Paper

Authors

1 Department of Mathematics, Science and Research branch, Islamic Azad University, Tehran, Iran

2 Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran

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 $G\cup W(S)$ 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 $3$ or $5$, chordal graph or $C_5$, then $G \cup W(S)$ is a vertex decomposable graph.

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