Advantage in the discrete Voronoi game

TL;DR

This paper analyzes the discrete Voronoi game on graphs, showing how the second player can dominate on some graphs but not on bounded-degree graphs, and providing bounds for trees.

Contribution

It establishes bounds for player outcomes in the discrete Voronoi game on various graph classes, including trees and bounded-degree graphs.

Findings

Second player can dominate on some graphs.

First player can secure at least 25% of vertices on trees.

Bounded-degree graphs limit the second player's advantage.

Abstract

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.