Upon further reflection, it seems although the direction of an edge is observable and is helpful in game play, it should not be a foundamental part of the game. In a graph world with higher valences there may indeed be no shared direction between vertices. The directed graph inspired by direction, however adds an interesting dimension to the game. Let us preserve that and restate the game:

The game environment(the game):

• Set of players P, typically players are distinguished by color. E.g. black and white.
• A total ordering O is placed on P to establish order of play with lesser player playing before greater player.
• Set of vertices V. Vertex can be empty or occupied by a single colored stone.
• Set of directed edges, E, between vertices : V X V.
• The freedom of a stone is sum of its individual liberties: A stone s receive one(1) liberty from any unoccupied vertex c if (c,s) is a directed edge. A stone located at s may benefit from freedom of a second stone, of the same color, located at t if (t,s) is a directed edge.

The game play

• The game as defined above is prepared.
• Players play by taking turns according to O.
• At each turn, a player may pass or placed a stone of his color onto an empty vertex.
• After stone placement the player chooses zero or more dead stones and removes them. A stone is dead if it has no freedom.
• After all stones are inspected, dead stones are removed.
• A stone cannot be placed to cause the board to repeat a situation previously seen in this game.
• Game terminated when all players pass consecutively.
• Player are ranked by number of stones on the board with their color. Winning player has more stone than losing player.
• Game winner has the most stones with his color on the board at the end of the game.