13 Prove that GRAPH-SEARCH satisfies the graph separation property illustrated in Figure 9. (Hint: Begin by showing that the property holds at the start, then show that if it holds before an iteration of the algorithm, it holds afterwards.) Describe a search algorithm that violates the property.

X 12 x 2 x 2 Figure 32 x 16 The track pieces in a wooden railway set; each is labeled with the number of copies in the set. Note that curved pieces and "fork" pieces ("switches" or "points") can be flipped over so they can curve in either direction. Each curve subtends 45 degrees.