  The graph grabbing game is played on a non-negatively weighted connected
graph by Alice and Bob who alternately claim a non-cut vertex from the
remaining graph, where Alice plays first, to maximize the weights on their
respective claimed vertices at the end of the game when all vertices have been
claimed. Seacrest and Seacrest conjectured that Alice can secure at least half
of the weight of every weighted connected bipartite even graph. Later, Egawa,
Enomoto and Matsumoto partially confirmed this conjecture by showing that Alice
wins the game on a class of weighted connected bipartite even graphs called
$K_{m,n}$-trees. We extend the result on this class to include a number of
graphs, e.g. even blow-ups of trees and cycles.

Combinatorics