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Graph Sharing Games: Complexity and Connectivity

  • Josef Cibulka
  • Jan Kynčl
  • Viola Mészáros
  • Rudolf Stolař
  • Pavel Valtr
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6108)

Abstract

We study the following combinatorial game played by two players, Alice and Bob. 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 under all the three combinations of rules (T) and (R), 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 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 variation of the game 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.

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References

  1. 1.
    Cibulka, J., Kynčl, J., Mészáros, V., Stolař, R., Valtr, P.: Solution of Peter Winkler’s Pizza Problem. In: Fete of Combinatorics. Springer, New York (to appear); Extended abstract in: Fiala, J., Kratochvíl, J., Miller, M. (eds.): IWOCA 2009. LNCS, vol. 5874, pp. 356–368. Springer, Heidelberg (2009)Google Scholar
  2. 2.
    Knauer, K., Micek, P., Ueckerdt, T.: How to eat 4/9 of a pizza (submitted)Google Scholar
  3. 3.
    Micek, P., Walczak, B.: Parity in graph sharing games (manuscript)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Josef Cibulka
    • 1
  • Jan Kynčl
    • 2
  • Viola Mészáros
    • 2
    • 3
  • Rudolf Stolař
    • 1
  • Pavel Valtr
    • 2
  1. 1.Department of Applied MathematicsCharles University, Faculty of Mathematics and PhysicsPraha 1Czech Republic
  2. 2.Department of Applied Mathematics and, Institute for Theoretical Computer ScienceCharles University, Faculty of Mathematics and PhysicsPraha 1Czech Republic
  3. 3.Bolyai InstituteUniversity of SzegedSzegedHungary

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