  Pebbling is a game played on a graph. The single player is given a graph and
a configuration of pebbles and may make pebbling moves by removing 2 pebbles
from one vertex and placing one at an adjacent vertex to eventually have one
pebble reach a predetermined vertex. The pebbling number, $\pi(G)$, is the
minimum number of pebbles such that regardless of their exact configuration,
the player can use pebbling moves to have a pebble reach any predetermined
vertex.
  Previous work has related $\pi(G)$ to the diameter of $G$. Clarke, Hochberg,
and Hurlbert demonstrated that every connected undirected graph on $n$ vertices
with diameter 2 has $\pi(G) = n$ unless it belongs to an exceptional family of
graphs, consisting of those that can be constructed in a specific manner; in
which case $\pi(G) = n +1$. By generalizing a result of Chan and Godbole,
Postle showed that for a graph with diameter $d$, $\pi(G) \le n 2^{\lceil
\frac{d}{2} \rceil} (1+o_n(1))$.
  In this article, we continue this study relating pebbling and diameter with a
focus on directed graphs. This leads to some surprising results. First, we show
that in an oriented directed graph $G$ (in the sense that if $i \to j$ then we
cannot have $j \to i$), it is indeed the case that if $G$ has diameter 2,
$\pi(G) = n$ or $n + 1$, and if $\pi(G) = n+1$, the directed graph has a very
particular structure. In the case of general directed graphs (that is, if $i
\to j$, we may or may not have an arc $j \to i$) with diameter 2, we show that
$\pi(G)$ can be as large as $\frac32 n + 1$, and further, this bound is sharp.
More generally, we show that for general directed graphs, $\pi(G) \le 2^d n / d
+ f(d)$ where $f(d)$ is some function of only $d$.
