Convex Grabbing Game of the Point Set on the Plane

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.