Given a (finite symmetric) graph $G$, consider the following 2-person game: The player who is to move chooses a vertex and removes it with all its neighbours. The player who is to move last wins. A graph is said to be trivial if the outcome is determined regardless of how the players move. The authors prove that a graph is trivial if and only if it belongs to ${\cal P}$, the collection of all graphs such that every maximal independent set is of the same parity. The family ${\cal P}$ is investigated; e.g., it is proved that if $G$ has no cycles of length $\le 7$, then $G\in{\cal P}$ if and only if every interior vertex has an odd number of leaves. In conclusion, a class of nontrivial graphs with the corresponding analyzable games is presented.