Partisan Combinatorial Game of Edge and Vertex Removal on Graphs

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.