A cage is a smallest undirected graph with the following two properties (for some constants $r,g$):
- Each node has exactly $r$ neighbors;
- The smallest cycle has exactly $g$ nodes.
When $r=2$ a cage is just a cycle with $g$ nodes, so cages usually become interesting only for $r\geq 3$.
I am interested in a "directed cage": a smallest directed graph with the following properties:
- Each node has exactly $r$ outgoing edges;
- The smallest undirected cycle (- a cycle in which the edge directions are ignored) has exactly $g$ nodes.
MY QUESTION IS: Is anything known about such "directed cages"?
Specifically: how many nodes are needed for a directed cage when $r=2$, as a function of $g$?