If you delete a node from a Dynkin diagram associated to a group $G$, you get a group which is a subgroup of $G$. In fact, it is a parabolic subgroup. (I can't come up with a reference for this, but I'm pretty sure that standard texts will state this fact. For example see Carter
Simple groups of Lie type or Wilson
The finite simple groups.) I don't know if this process has a name, it is a well known fact in the theory of (finite and algebraic) groups of Lie type, though.
For example you have $$A_1<A_2<...<A_7<E_8$$
As further information, check the wikipedia page on root datum
A connected split reductive algebraic group over K is uniquely
determined (up to isomorphism) by its root datum, which is always
reduced. Conversely for any root datum there is a reductive algebraic
group. A root datum contains slightly more information than the Dynkin
diagram, because it also determines the center of the group.