On Bichromatic Triangle Game

We study a combinatorial game called Bichromatic Triangle Game, defined as follows. Two players RR and BB construct a triangulation on a given planar point set  VV. Starting from no edges, they take turns drawing one straight edge that connects two points in VV and does not cross any of the previously drawn edges. Player RR uses color red and player BB uses color blue. The first player who completes one empty monochromatic triangle is the winner. We show that each of the players can force a tie in the Bichromatic Triangle Game when the points of VV are in convex position, and also in the case when there is exactly one inner point in the set  VV.As a consequence of those results, we obtain that the outcome of the Bichromatic Complete Triangulation Game (a modification of the Bichromatic Triangle Game) is also a tie for the same two cases regarding the set  VV.