Graph sharing games: Complexity and connectivity

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 ε > 0 and for every k ≥ 1 there is a k-connected graph G for which Bob has a strategy to obtain (1 − ε) of the total weight of the vertices. This contrasts with the original Pizza game played on a cycle, where Alice is known to have a strategy to obtain 4/9 of the total weight.We show that the problem of deciding whether Alice has a winning strategy (i.e., a strategy to obtain more than half of the total weight) is PSPACE-complete if condition (R) or both conditions (T) and (R) are required. We also consider a game played on connected graphs (without weights) where the first player who violates condition (T) or (R) loses the game. We show that deciding who has the winning strategy is PSPACE-complete.

Graph Sharing Game and the Structure of Weighted Graphs with a Forbidden Subdivision

“…Whether the same is true for game T is open. Cibulka et al [4] also asked about the complexity of finding an optimal strategy in games T and R on trees. A polynomial-time algorithm for game T on trees was devised by Walczak [20].…”

“…The above is one of the two graph sharing games introduced by Cibulka, Kynčl, Mészáros, Stolař, and Valtr [4] and independently by Micek and Walczak [13,14]. They called it the graph sharing game with taken part connected or game T .…”

“…Easy examples show that there is no hope in obtaining a similar result for either game T or game R on general graphs [4,13,14]. However, an appropriate restriction on the parity of the number of vertices and the structure of the graph can yield existence of good strategies of Alice.…”

Extremal problems on planar point sets

“…For any of the mentioned three rules and for any k ≥ 1 we present a k-connected graph where Bob has a strategy to obtain almost all of the total weight. We also prove the PSPACE-completeness of some decision problems formulated in this context [8].…”

“…In a follow-up paper we investigate generalizations of this game [8]. Instead of a cycle any graph G can be considered.…”