I am studying the branching structure of a Directed Acyclic Graph and need to define a kind of set that includes all the possible reachable nodes according to the branching path.
For example, I have a graph that consists of three nodes $A$, $B$, and $C$, with directed paths $A\rightarrow B\rightarrow C$ and $A\rightarrow C$, which is a triangular structure. I want to define a set of branches originating from A, such as $S(A)=\{A, AB, AC, ABC, ABC^2\}$, explanations are as follows:
- $A$: $A\nrightarrow $, starting at $A$, but no child nodes are reached.
- $AB$: $A\rightarrow B$ and $A\nrightarrow C$, starting at $A$ and reaching $B$, not reaching $C$.
- $AC$: $A\rightarrow C$ and $A\nrightarrow B$, reaching $C$, not reaching $B$.
- $ABC$: $A\rightarrow B\rightarrow C$, or $A\rightarrow B\nrightarrow C$ and $A\rightarrow C$, reaching both $B$ and $C$.
- $ABC^2$: Traversing $A\rightarrow B\rightarrow C$ and $A\rightarrow C$, representing two occurrences of reaching C.
Are there established definitions for such structures, and can you provide insights or references? I am unsure if this representation is rigorously accurate. If any discrepancies exist, I apologize and kindly request guidance in refining the terminology.