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 paper we examine the distance game for various graph operations: the join, the corona and the lexicographic product of graphs. We provide general bounds and exact results for special graphs
Cáceres, J. [et al.]. The graph distance game and some graph operations. "Electronic notes in discrete mathematics", 02 Setembre 2014, vol. 46, p. 153-159.