Let $G$ be an undirected graph.


* The eccentricity of a vertex $v$ of $G$, is the maximum distance between $v$ and any other vertex of $G$:$\;\;$ $\mathit{ecc}(v) = \max\_{u}\mathit{dist}(v,u)$.
* The radius of $G$ is the minimum eccentricity of a vertex of $G$:$\;\;$$R(G) = \min\_v \mathit{ecc}(v)$.
* The diameter of $G$ is the maximum eccentricity of a vertex of $G$: $\;\;$ $D(G) = \max\_v \mathit{ecc}(v)$.


**Question:** Clearly we have $R(G)\leq D(G)$. Is it possible to improve this upper-bound of $R(G)$ in function of $D(G)$ on maximal planar graphs? For instance, if $G$ is *outerplanar*, then it is known that $R(G)\leq \lfloor\frac{D(G)}{2}\rfloor + 1$. Is the same relation valid for maximal planar graphs? 


Obs1: An outerplanar graph is a planar graph which can be drawn in the plane in such a way that no two edges cross and all vertices belong to the outer-face of the drawing.


Obs2: A maximal planar (outerplanar) graph is a planar (outerplanar) graph whose addition of one edge destroys the property of being planar (outerplanar). 



Diameter vs Radius in Maximal Planar Graphs
The inequality
$$R(G)\leq \lfloor\frac{D(G)}{2}\rfloor + 1$$
does not hold in general for maximal planar graphs. For a counterexample, let $G$ be the icosahedral graph. Every face of the icosahedron is a triangle, and a planar graph is maximal if and only if it is a triangulation, so $G$ is a maximal planar graph. The radius and diameter of $G$ are both $3$, but the inequality above would require $R(G) \leq 2$.


I don't know what the best general upper bound is for maximal planar graphs.


