Complexity of critical positions I understand that chess is pretty complex regarding the game tree complexity (Shannon number) or number of possible positions. If we were trying to solve chess (whuch I know is impossible for practical reasons), however we might not be interested in all positions. For instance I remember reading that for the 7 men tablebase they excluded positions where one side had a bare king.

Is there an estimate for the complexity of chess if clearly won positions are excluded?
The problem every time you have such enormous number of configurations, is that any "simple-minded" description of configurations to discard will remove an insignificant portion of the total.

In the case at hand, there are trillions upon trillions of positions with a bare king (for instance), but they represent a small fraction of a percent of all positions. 

You still have to deal with the bulk of configurations that make chess hard, so the difficulty of the original problem is still there.