  The diameter of an undirected or a directed graph is defined to be the
maximum shortest path distance over all pairs of vertices in the graph. Given
an undirected graph $G$, we examine the problem of assigning directions to each
edge of $G$ such that the diameter of the resulting oriented graph is
minimized. The minimum diameter over all strongly connected orientations is
called the oriented diameter of $G$. The problem of determining the oriented
diameter of a graph is known to be NP-hard, but the time-complexity question is
open for planar graphs. In this paper we compute the exact value of the
oriented diameter for triangular grid graphs. We then prove an $n/3$ lower
bound and an $n/2+O(\sqrt{n})$ upper bound on the oriented diameter of planar
triangulations. It is known that given a planar graph $G$ with bounded
treewidth and a fixed positive integer $k$, one can determine in linear time
whether the oriented diameter of $G$ is at most $k$. In contrast, we consider a
weighted version of the oriented diameter problem and show it to be is weakly
NP-complete for planar graphs with bounded pathwidth.
