The Game of Take Turn

We introduce the game of Take Turn, a vertex-deletion game played with coins placed on vertices of a graph. A move is to remove a heads-up coin and turn over any coins adjacent from the removed coin; the loser is the player who cannot remove a coin. We relate the Sprague-Grundy values for games played on directed paths and cycles to the octal game .37 and examine cases for games played on undirected paths. We also show that Take Turn played on directed graphs is PSPACE-Complete.