  The concept of metric dimension has applications in a variety of fields, such
as chemistry, robotic navigation, and combinatorial optimization. We show
bounds for graphs with $n$ vertices and metric dimension $\beta$. For
Hamiltonian outerplanar graphs, we have $\beta \leq
\left\lceil\frac{n}2\right\rceil$; for outerplanar graphs in general, we have
$\beta \leq \left\lfloor\frac{2n}{3}\right\rfloor$; for maximal planar graphs,
we have $\beta \leq \left\lfloor\frac{3n}{4}\right\rfloor$. We also show that
bipyramids have a metric dimension of $\left\lfloor\frac{2n}{5}\right\rfloor +
1$. It is conjectured that the metric dimension of maximal planar graphs in
general is on the order of $\left\lfloor\frac{2n}{5}\right\rfloor$.
