In the graph distance game, two players alternate in constructing a maximal path. The objective function is the distance between the two endpoints of the path, which one player tries to maximize and the other tries to minimize. In this note, we examine the distance game for various graphs, and provide general bounds, exact results for special graphs, and an algorithm for trees. Computer calculations suggest interesting conjectures for grids.
Goddard W, Sinko A, Slater PJ, Xu H. 2011. The graph distance game. AKCE International Journal of Graphs and Combinatorics 8(1): 85-96.