  In 2020, Cameron et al. introduced the restricted numerical range of a
digraph (directed graph) as a tool for characterizing digraphs and studying
their algebraic connectivity. In particular, digraphs with a restricted
numerical range of a single point, a horizontal line segment, and a vertical
line segment were characterized as $k$-imploding stars, directed joins of
bidirectional digraphs, and regular tournaments, respectively. In this article,
we extend these results by investigating digraphs whose restricted numerical
range is a convex polygon in the complex plane. We provide computational
methods for identifying these polygonal digraphs and show that these digraphs
can be broken into three disjoint classes: normal, restricted-normal, and
pseudo-normal digraphs, all of which are closed under the digraph complement.
We prove sufficient conditions for normal digraphs and show that the directed
join of two normal digraphs results in a restricted-normal digraph. Also, we
prove that directed joins are the only restricted-normal digraphs when the
order is square-free or twice a square-free number. Finally, we provide methods
to construct restricted-normal digraphs that are not directed joins for all
orders that are neither square-free nor twice a square-free number.
