This question is taken from Category Theory for Programmers by Bartosz Milewski (1.6).
Q: When is a directed graph a category?
My intuition is that for a directed graph $G$ to have category structure, the existence of edges (arrows) $A \rightarrow B$ and $B \rightarrow C$ would imply the existence of the edge/arrow $A \rightarrow C$. I'm not quite sure if that point is correct, or if the meaning is something weaker like $A \rightarrow B \rightarrow C\rightarrow D, D \rightarrow B \implies D \rightarrow C$
Assuming the stronger condition, That would imply that any pair of connected vertices must have an arrow directly linking them, which would imply that $G$, viewed as an undirected graph, would have the same connectivity as a graph made of complete graphs with some overlapping vertices. More precisely, any connected subgraph $H \subset G$, $|H| = n$, when viewed as an undirected graph, must have $ H \cong K_n$. Can something stronger be said?