  We study the discrete Voronoi game, where two players alternately claim
vertices of a graph for t rounds. In the end, the remaining vertices are
divided such that each player receives the vertices that are closer to his or
her claimed vertices. We prove that there are graphs for which the second
player gets almost all vertices in this game, but this is not possible for
bounded-degree graphs. For trees, the first player can get at least one quarter
of the vertices, and we give examples where she can get only little more than
one third of them. We make some general observations, relating the result with
many rounds to the result for the one-round game on the same graph.
