Perhaps a bit more formally, is there a name for regions delimited by nodes A and B, in a directed graph, where all paths starting from A, when prolonged, will eventually reach B, and all paths reaching B are crossing A, or start from a node that is reachable from A?
EDIT: Hopefully more precisely, thanks to the comment:
I am interested in subgraphs $H(A,B)$ of $G$, with $A \in G, B \in G$:
- $\forall v \in H$, $v$ is reachable from $A$,
- $\forall v \in H$, $B$ is reachable from $v$,
- $\forall x \in G, x \notin H, v \in H$, if $v$ is reachable from $x$, then all paths between $x$ and $v$ must cross $A$,
- $\forall x \in G, x \notin H, v \in H$, if $x$ is reachable from $v$, then all paths between $v$ and $x$ must cross $B$.
Given a node $A$, there can be several valid subgraphs $H(A,B)$. In the set of all valid subgraphs, I am interested in the one that minimizes the distance between $A$ and $B$ (I believe it unique in my use case).
In This graph, the following graphs are examples of what I am looking for:
- $H(001,160)$, containing nodes $001, 160, 273$;
- $H(160,300)$, containing nodes $160, 255, 203, 016, 048, 133, 268, 348, 333, 125, 067, 300$
The subgraph between node $160$ and $333$ doesn't fit the bill.