  Given a simple polygon $P$ and a set $Q$ of points contained in $P$, we
consider the geodesic $k$-center problem where we want to find $k$ points,
called \emph{centers}, in $P$ to minimize the maximum geodesic distance of any
point of $Q$ to its closest center. In this paper, we focus on the case for
$k=2$ and present the first exact algorithm that efficiently computes an
optimal $2$-center of $Q$ with respect to the geodesic distance in $P$.
