Cut the Board

Suzumo has a chessboard with $N$ rows and $M$ columns. In one step, he can choose two cells of the chessboard which share a common edge (that has not been cut yet) and cut this edge. Formally, the chessboard is *split* into two or more pieces if it is possible to partition its cells into two non-empty subsets $S_1$ and $S_2$ ($S_1\cap S_2 = \emptyset$, $|S_1|+|S_2|=NM$) such that there is no pair