The graph grabbing game on $\{0,1\}$-weighted graphs

The \emph{graph grabbing game} is a two-player game on a weighted connected graph in which two players, Alice and Bob, alternatively remove non-cut vertices one by one to gain the weights on them. Alice wins the game if she gains at least half of the total weights. In this paper, we show that on every connected even graph which does contain a fully spiked cycle as an induced subgraph, Alice always has a winning strategy with an arbitrary weight function whose codomain is $\{0,1\}$. In addition, we give a list of forbidden subgraph for the family of graphs on which Alice has a winning strategy with an arbitrary weight function whose codomain is $\{0,1\}$.