Grids containment and treewidth , PTASs

Definition 1. A graph G is a t × t grid if it consists of t 2 vertices forming a square t × t such that every two consecutive vertices in a row or in a colum are connected. As a result of previous excercises we know that a treewidth of a t × t grid is t. A treewidth of a graph can only decrease as a result of taking minors so we can conclude that if a graph contain a t × t grid as a minor it's treewidth is at least t. Following series of theorems gives us a symetric dependence.