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Graphs and Combinatorics

, Volume 36, Issue 1, pp 51–62 | Cite as

Convex Grabbing Game of the Point Set on the Plane

  • Naoki MatsumotoEmail author
  • Tomoki Nakamigawa
  • Tadashi Sakuma
Original Paper

Abstract

We introduce a new combinatorial game of a weighted point set P on the plane in general position, called a convex grabbing game. In the game, two players alternately remove a point on the convex hull of P and obtain the weight of the removed point as their score. Each player’s aim is to maximize their score, when all points have been taken. In this paper, we prove that the first player can always win the game on the given point set of odd points with at most two inner points. Moreover, by restricting the weight of each point to zero or one, we relax the condition “at most two” in the above result to “at most four”. We also show that these results are best possible by constructing several weighted point sets in which the first player cannot win the game.

Keywords

Convex grabbing game Graph grabbing game Convex geometry 

Notes

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Copyright information

© Springer Japan KK, part of Springer Nature 2020

Authors and Affiliations

  • Naoki Matsumoto
    • 1
    Email author
  • Tomoki Nakamigawa
    • 2
  • Tadashi Sakuma
    • 3
  1. 1.Research Institute for Digital Media and ContentKeio UniversityYokohamaJapan
  2. 2.Shonan Institute of TechnologyFujisawaJapan
  3. 3.Faculty of ScienceYamagata UniversityYamagataJapan

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