Lipton's most influential results Richard J. Lipton has been selected as the winner of the 2014 Knuth Prize  "for Introduction of New Ideas and Techniques". 

What are to your minds the main new ideas and techniques that Lipton developed?

Note. This question shall become community wiki, please put one such idea, technique or result per answer.
The Planar Separator Theorem states that in any planar $n$-vertex graph $G$ there exists a set of $O(\sqrt{n})$ vertices whose removal leaves the graph disconnected into at least two roughly balanced components. Moreover, such a set can be found in linear time. This (tight) result, proved by Lipton and Tarjan (improving on a previous result by Ungar) is a powerful tool for designing algorithms on planar graphs. It gives many exact subexponential time algorithms for NP-hard problems and improved polynomial time approximation algorithms. Looking at the wikipedia page gives a good starting place to explore the numerous applications. An early survey with details of a number of applications was written by Lipton and Tarjan in 1980.