Graph sharing games: complexity and connectivity
Josef Cibulka, Jan Kyn\v{c}l, Viola M\'esz\'aros, Rudolf Stola\v{r},, Pavel Valtr

TL;DR
This paper explores the complexity of a two-player graph sharing game, demonstrating that certain variants are PSPACE-complete and that players can secure nearly all the vertices' weight on highly connected graphs.
Contribution
It introduces a generalized graph sharing game, analyzes its complexity, and establishes PSPACE-completeness results for various rule conditions and game variants.
Findings
Deciding if Alice can win is PSPACE-complete under certain rules.
High connectivity graphs allow Bob to secure almost all vertices.
The game generalizes the Pizza game and exhibits complex strategic behavior.
Abstract
We study the following combinatorial game played by two players, Alice and Bob, which generalizes the Pizza game considered by Brown, Winkler and others. Given a connected graph G with nonnegative weights assigned to its vertices, the players alternately take one vertex of G in each turn. The first turn is Alice's. The vertices are to be taken according to one (or both) of the following two rules: (T) the subgraph of G induced by the taken vertices is connected during the whole game, (R) the subgraph of G induced by the remaining vertices is connected during the whole game. We show that if rules (T) and/or (R) are required then for every epsilon > 0 and for every positive integer k there is a k-connected graph G for which Bob has a strategy to obtain (1-epsilon) of the total weight of the vertices. This contrasts with the original Pizza game played on a cycle, where Alice is known to…
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