Extrema property of the kk-ranking of directed paths and cycles

A kk-ranking of a directed graph GG is a labeling of the vertex set of GG with kk 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     GG is defined to be the smallest kk such that GG has a kk-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 Sierpiński 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.