Bounds on metric dimension for families of planar graphs

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 $\left\lfloor\frac{2n}{5}\right\rfloor$.