The graph grabbing game on blow-ups of trees and cycles

TL;DR

This paper studies a strategic graph game played on weighted graphs, proving that Alice can secure a significant portion of the total weight on certain classes of graphs, including blow-ups of trees and cycles.

Contribution

It extends previous results by demonstrating Alice's winning strategy on broader classes of graphs such as blow-ups of trees and cycles.

Findings

Alice can secure at least half the weight on certain bipartite graphs.

The result applies to even blow-ups of trees and cycles.

The paper generalizes previous conjectures to new graph classes.

Abstract

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.