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

TL;DR

This paper studies the graph grabbing game on weighted graphs, showing conditions under which Alice can guarantee a win with simple binary weights and identifying forbidden subgraphs for winning strategies.

Contribution

It characterizes graphs where Alice has a winning strategy with binary weights, including the role of fully spiked cycles and forbidden subgraphs.

Findings

Alice has a winning strategy on certain even graphs with fully spiked cycles.

A list of forbidden subgraphs for winning strategies with binary weights is provided.

Conditions for guaranteed wins in the graph grabbing game are established.

Abstract

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 $\{0,1\}$.