# What do you call a set of vertices that separates the root from the leaves?

Suppose we are given a rooted tree $$T$$, and a set of vertices $$M$$ that separates the root of $$T$$ from its leaves. In other words, every path from the root of $$T$$ to a leaf contains a vertex in $$M$$. Is there a standard term for such a set $$M$$?

• If you want to keep the arborial metaphor going, perhaps "complete pruning" would be appropriate. – Ryan Budney Mar 17 at 2:14
• If this doesn't have a name yet, I recommend calling it a topping. – Jesko Hüttenhain Mar 17 at 2:21
• How about "non-leaves"? – David G. Stork Mar 21 at 0:25
• @DavidG.Stork The set $M$ may contain leaves. For example, note that if we choose $M$ to be the set of all leaves, then it indeed satisfies the property that every path from the root to a leaf passes through $M$. – Or Meir Mar 21 at 13:33
• Define "through." – David G. Stork Mar 21 at 13:46

Call the root node $$r$$. If you join each leaf node to a dummy sink node $$t$$, then $$M$$ would be an $$(r,t)$$ vertex separator, also known as vertex cut or separating set.
Even without the dummy node, $$M$$ is an $$(r,\ell)$$ vertex separator for each leaf node $$\ell$$.
• ? What about honoring $M$ ?! – Wlod AA Mar 17 at 3:54