Proximity and Radius in Outerplanar Graphs with Bounded Faces

Let $G$ be a finite, connected graph and $v$ a vertex of $G$. The average distance and the eccentricity of $v$ in $G$ are defined as the arithmetic mean and the maximum, respectively, of the distances from $v$ to all other vertices of $G$. The proximity of $G$ and the radius of $G$ are defined as the minimum of the average distances and the eccentricities over all vertices of $G$. In this paper, we establish an upper bound on the proximity of a $2$-connected outerplanar graphs in terms of order and maximum face length. This bound is sharp apart from a small additive constant. It is known that the radius of a maximal outerplanar graph is at most $\lfloor \frac{n}{4} \rfloor +1$. In the second part of this paper we show that this bound on the radius holds for a much larger subclass of outerplanar graphs, for all $2$-connected outerplanar graphs of order $n$ whose maximum face length does not exceed $\frac{n+2}{4}$.