I have a directed graph with two sets, vertices and edges, $G=(V,E)$.
What is the mathematical correct notation for all the successors / predecessors of a vertex?
At the moment, for the direct successors I use the out-neighborhood $N^+(v)$ or simply $S_v$. Equivalent for the direct predecessors. Is this a good way to do this? And what is the way to do it, if I want also the successors' successors (and so on) in the result set?
Best Answer
Yes, that is out I would refer to the successors of $v$ as $N^{+}(v)$ and the predecessors of $v$ as $N^{-}(v)$. After all, successors is just a different word for out-neighborhood.
For the successors of the successors, I would use $N^{+}(N^{+}(v))$. You could shorthand it to to ${N^{+}}^{2}(v)$ or a different notation you define.
Edit: For the set of all successors of $v$ in graph $G$, i.e. the set of all vertices reachable from $v$, I've seen this set denoted as $R_{G}(v)$ or $R_{G}^{+}(v)$. The set of predecessors, i.e. the set of vertices that can reach $v$, is denoted $R_{G}^{-}(v)$.