The game of removing two vertices in a graph Consider the following impartial combinatorial game played with finite graphs: A move removes two adjacent vertices; and of course all edges connected with them. The game then continues with the new graph. Often this turns out to be disconnected, so we end up in a sum of smaller games. Let us choose the normal play rule (if no move is possible, one loses).
Question. Does this game have a name? Has it been studied anywhere?
It seems to me that Cram is a special case of this game: To each box corresponds a vertex. These are adjacent via some edge iff the boxes are adjacent. This is also mentioned in the 2nd Volume of Winning Ways (by Berlekamp, Guy, Conway), where Cram is turned directly into a game on graphs. The authors mention that we can play arbitrary graphs, but only study grid-like graphs which arise in Cram. The game is also mentioned in the paper Let us play with qubits in section 3.2., denoted "the classical domino game" on graphs, but also restricts to grid-like graphs.
 A: Per Martin's request, here is a more detailed version of my comment.  Given a graph $G$, the independence game is the game in which two players take turn in removing a vertex and all of its neighbors.  The game terminates when a player can make no further move, and it is a win for the player who last moved in the normal play and for the other one in the misère play.
The game that Martin suggests with starting graph $G$ is the special case of the independence game, played on the line dual of $G$: the line dual of a graph $G$ is the graph whose vertices are the edges of $G$ and whose edges are pairs of non-disjoint edges of $G$.
In particular, when the initial graph is a path, its line dual is also a path and the game is the same as what is called Dawson's chess.  In this case, the independence game on a path with $n$ vertices with normal play, is a second player win if and only if either $n  \in \{ 0,  14,  34\}$ or $n \equiv 4,  8,  20,  24,  28 \pmod{34}$.
A: If you haven't already, take a look at "Kayles on the way to the stars" by R. Fleischer, G. Trippen.
It is a paper about the analysis of the game of Kayles on graphs, which shouldn't be far from Dawson's Kayles on graphs, which is similar to what you're looking for.
Please note that the authors adopted the rule of removing nodes rather than edges.
