This is because there are 2ℓ i edges within V i at the left and right boundaries of the ℓ i spans, and at least t 2 − t 1 edges connecting the extra vertices from T 2 that have no matching vertex in T 1 .
from assumption (5) we have which in turn implies Similarly, using (6), we have That is, for T to have small edge expansion, the vertices in T 1 and T 2 must be collected into a small number of spans ℓ 1 and ℓ 2 .