The RAAGs on the complement graphs of path graphs in mapping class groups

In this article, we determine the function $\ell(S_{g, p})$ such that the right-angled Artin group $G(P_{m})$ is embedded in the mapping class group $\mathrm{Mod}(S_{g, p})$ if and only if $m$ is not more than $\ell(S_{g, p})$. Using this function and Birman--Hilden theory, we prove that $\mathrm{Mod}(S_{0, p})$ is virtually embedded in $\mathrm{Mod}(S_{g, 0})$ if and only if $p \leq 2g+2$.