  Let $P$ be a path graph of $n$ vertices embedded in a metric space. We
consider the problem of adding a new edge to $P$ so that the radius of the
resulting graph is minimized, where any center is constrained to be one of the
vertices of $P$. Previously, the "continuous" version of the problem where a
center may be a point in the interior of an edge of the graph was studied and a
linear-time algorithm was known. Our "discrete" version of the problem has not
been studied before. We present a linear-time algorithm for the problem.
