Given a series parallel graph $G$ with sink $t$ let $v$ be a node of $G$ that is not $t$. Call a node $w$ of $G$ a bottleneck if each $v$-$t$ (directed) path in $G$ goes through $w$. Let $w_1,w_2$ be two distinct lower bottlenecks of $v$. Let $P_{v,t}$ be a $v$-$t$ (directed) path. $P_{v,t}$ intersects both $w_1$ and $w_2$. W.l.o.g $P_{v,t}$ intersects $w_1$ first. So $w_1$ is an ancestor of $w_2$. Thus any two distinct lower bottlenecks of $v$ are in an ancestor descendant relationship. Then one can talk about a highest bottleneck of $v$.
Has this notion been defined before and is there a standard terminology for this?
Here edges of our series parallel graph are directed from source to sink.
1 Answer 1
$v$ is trivially on every path from $v$ to $t$, but presumably that isn't what you're looking for. The "highest bottleneck" is the meet of the set of direct successors of $v$, as defined here.