  We consider three variants of a partisan combinatorial game between two
players, Left and Right, played on an undirected simple graph. Left is able to
delete vertices (and incident edges) while Right is able to delete edges. This
natural extension of a similar impartial game gives a clear advantage to one
player by allowing them the ability to play on a small subgraph which the other
player can not. Our last variant removes this advantage by assuming a move is
valid for one player if and only if the other player has a valid move on the
same graph. In this case, we show that the ability to remove a vertex is more
advantageous compared to removing edges.
