Compute visible vertices of a polygon

Question

Given a simple polygon $P$ and a point $p$ within the polygon, I want to compute the vertices of $P$ visible from point $p$. A point $q$ is visible from point $p$, if the line segment $\overline{qp}$ is completely in the polygon $P$.

One can use the accepted answer here to compute visible vertices in $P$ from a point $p$ in $\mathcal{O}(n\log{}n)$.

My question is, does an algorithm exist to do it in $\mathcal{O}(n)$ time?