Right-angled Artin groups and full subgraphs of graphs

Abstract: For a finite graph $\Gamma$, let $G(\Gamma)$ be the right-angled Artin group defined by the complement graph of $\Gamma$. We show that, for any linear forest $\Lambda$ and any finite graph $\Gamma$, $G(\Lambda)$ can be embedded into $G(\Gamma)$ if and only if $\Lambda$ can be realised as a full subgraph of $\Gamma$. We also prove that if we drop the assumption that $\Lambda$ is a linear forest, then the above assertion does not hold, namely, for any finite graph $\Lambda$, which is not a linear forest, there exists a finite graph $\Gamma$ such that $G(\Lambda)$ can be embedded into $G(\Gamma)$, though $\Lambda$ cannot be embedded into $\Gamma$ as a full subgraph.