I have some additional results on the cop and killer game, with the intent of being able to characterize outcomes using retracts of the original graphs. Because I used some features of LaTeX not available on AoPS (such as defining some of the symbols I use), I'm just attaching the PDF here for now; if I have time later I'll convert to inline BBCode and update this post.
Let be a connected region in the plane. Suppose that the cop is a segment of length and the robber is a segment of length .
The cop starts inside (so we assume that a segment of length fits somewhere inside ), as does the robber. Based on ,,, and the initial positions, does the cop catch the robber if both use optimal play?
A good place to start might be regions where all sides are parallel to the coordinate axes.
How does the result change if the cop and robber are circles or squares instead of segments?
What if the game is played inside connected regions in space?