The original coin-removal problem is given in D'Angelo and West:
Suppose that n coins are arranged in a a row. We remove heads-up coins, one by one. Each time we remove a coin we must flip the coins still present in the (at most) two positions surrounding it. For which arrangements of heads and tails can we remove all the coins?The problem was posed in Math 201; Michael Siler presented a solution to the HRS Symposium. We then went on to consider some variations of the problem, discussed in Variations of a Coin-Removal Problem.
However, the variations considered were either linear or circular (coins arranged in a line or a circle). Even more general would be to consider the coins arranged on the vertices of an arbitrary graph. It seems unlikely that a general solution for arbitrary graphs exists, but for specific classes of graphs (e.g., rectangular), a solution may be possible.
One of the circular variations was posed as a problem in the American Mathematical Monthly. Two solutions are given in the Nov. 1997 issue (problem #10459, vol. 104, no. 9, p. 876). A Java applet to play solitaire on the circle can be found at Cut the Knot.
Adam Isom put together a Java applet that plays the solitaire game on different graphs. Examples are: