= Rooted separation of a graph
= Rooted separations of a graph
{synonym}
For a specified <set> $S$ of <vertices>, a rooted separation is a pair $(A,B)$ with $A\cup B=V(G)$, $S\subseteq A$, and no <edges> between $A\setminus B$ and $B\setminus A$. Its order is $|A\cap B|$. It avoids a branch set $C$ when $A\cap C=\varnothing$. A graph with <vertex connectivity> at least $|S|$ has no rooted separation of order below $|S|$ avoiding a nonempty branch set: both sides of such a separation would be nonempty after deleting its separator.
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