I thought up the following two player game on a graph.
A graph $G$ is given as an input. Each player deletes one edge in turn. The one who makes an isolated vertex first wins.
I would like to know how to decide the winner and his strategy.
It is easy to see that the vertices with degree $1$ or $2$ are not needed. But I don't know further about this game even for simple graphs.
I would appreciate it if you would give me some observations or comments or related works or whatever.