This is a joint work with Takuya Katayama. Let $\Gamma$ be a finite graph without loops or multi-edges. In 2015, Kim-Koberda proved that for any $\Gamma$ there exists a positive integer $p$ such that the right-angled Artin group $A(\Gamma)$ on $\Gamma$ is a subgroup of the pure braid group $PB_{p}$ on $p$-strands. Our main concern is to decide whether $A(\Gamma)$ is a subgroup of $PB_{p}$, the braid group $B_{p}$ on $p$-strands, and more generally, the mapping class group of an orientable surface for given $\Gamma$. In this talk, we solve this problem for the complement graphs of path graphs.