  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 $\{0,1\}$.
