Embeddability of right-angled Artin groups on complements of trees

Abstract: For a finite simplicial graph $\Gamma$, let $A(\Gamma)$ denote the right-angled Artin group on $\Gamma$. Recently Kim and Koberda introduced the extension graph $\Gamma^e$ for $\Gamma$, and established the Extension Graph Theorem: for finite simplicial graphs $\Gamma_1$ and $\Gamma_2$ if $\Gamma_1$ embeds into $\Gamma_2^e$ as an induced subgraph then $A(\Gamma_1)$ embeds into $A(\Gamma_2)$. In this article we show that the converse of this theorem does not hold for the case $\Gamma_1$ is the complement of a tree and for the case $\Gamma_2$ is the complement of a path graph.