Covering Regions with Squares

A unit square in $R^2 $ whose corners are integer lattice points is called a block. A board consists of a finite set of blocks. Given a board $B$, its graph $G(B)$ has vertices corresponding with the blocks of $B$, and two vertices of $G(B)$ are joined by an edge provided the corresponding blocks are contained in a square subset of $B$. If $B$ is simply connected, then $G(B)$ is perfect.