  A $k$-ranking of a directed graph $G$ is a labeling of the vertex set of $G$
with $k$ positive integers such that every directed path connecting two
vertices with the same label includes a vertex with a larger label in between.
The rank number of $G$ is defined to be the smallest $k$ such that $G$ has a
$k$-ranking. We find the largest possible directed graph that can be obtained
from a directed path or a directed cycle by attaching new edges to the vertices
such that the new graphs have the same rank number as the original graphs. The
adjacency matrix of the resulting graph is embedded in the Sierpi\'nski
triangle.
  We present a connection between the number of edges that can be added to
paths and the Stirling numbers of the second kind. These results are
generalized to create directed graphs which are unions of directed paths and
directed cycles that maintain the rank number of a base graph of a directed
path or a directed cycle.
