I have a couple of questions about searching in graphs/trees:

Let's assume I have an empty chess board and I want to move a pawn around from point A to B.

A. When using depth first search or breadth first search must we use open and closed lists ? This is, a list that has all the elements to check, and other with all other elements that were already checked? Is it even possible to do it without having those lists? What about A*, does it need it?

B. When using lists, after having found a solution, how can you get the sequence of states from A to B? I assume when you have items in the open and closed list, instead of just having the (x, y) states, you have an "extended state" formed with (x, y, parent_of_this_node) ?

C. State A has 4 possible moves (right, left, up, down). If I do as first move left, should I let it in the next state come back to the original state? This, is, do the "right" move? If not, must I transverse the search tree every time to check which states I've been to?

D. When I see a state in the tree where I've already been, should I just ignore it, as I know it's a dead end? I guess to do this I'd have to always keep the list of visited states, right?

E. Is there any difference between search trees and graphs? Are they just different ways to look at the same thing?