Bandwidth reduction in rectangular grids

We show that the bandwidth of a square two-dimensional grid of arbitrary size can be reduced if two (but not less than two) edges are deleted. The two deleted edges may not be chosen arbitrarily, but they may be chosen to share a common endpoint or to be non-adjacent.
We also show that the bandwidth of the rectangular $n \times m$ ($n\leq m$) grid can be reduced by $k$, for all $k$ that are sufficiently small, if $m-n+2k$ edges are deleted.