It is interesting to have a new model for graphs om impartial combinatorial games. Most graph representations so describes the kernel of the game graph as the unique set of P-positions. I am not yet familiar with research on hte optimal game graph, but like to learn more. Particularly interesting is the connection to CA. We have a recent discovery of a CA representing the P-positions of Blocking Wythoff Nim: https://arxiv.org/abs/1506.01431

So what else is known regarding Wythoff Nim and consecutive P-positions in an optimal longest game?

There are many variations of Wythoff Nim, for example Holladays $k$ Wythoff nim, but also Maharaja Nim (the Queen can also move like a knight of Chess) and the GDWN family, where new `diagonals’ of rational slopes are adjoined as moves to Wythoff Nim. Each variation has its Evolution graph, and it would be interesting to understand them, but most of these game have non-well-behaved P-positions…

]]>