  An oriented graph has weak diameter at most $d$ if every non-adjacent pair of
vertices are connected by a directed $d$-path. The function $f_d(n)$ denotes
the minimum number of arcs in an oriented graph on $n$ vertices having weak
diameter $d$. Finding the exact value of $f_d(n)$ is a challenging problem even
for $d = 2$. This function was introduced by Katona and Szemeredi (1967), and
after that several attempts were made to find its exact value by Znam (1970),
Dawes and Meijer (1987), Furedi, Horak, Pareek and Zhu (1998), and Kostochka,
Luczak, Simonyi and Sopena (1999) through improving its best known bounds. In
that process, it was proved that this function is asymptotically equal to
$n\log_2 n$ and hence, is an asymptotically increasing function. However, the
exact value and behaviour of this function was not known.
  In this article, we observe that the oriented graphs with weak diameter at
most $2$ are precisely the absolute oriented cliques, that is, analogues of
cliques for oriented graphs in the context of oriented coloring. Through
studying arc-minimal absolute oriented cliques we prove that $f_2(n)$ is a
strictly increasing function. Furthermore, we improve the best known upper
bound of $f_2(n)$ and conjecture that our upper bound is tight. This
improvement of the upper bound improves known bounds involving the oriented
achromatic number.
