  The Explorer-Director game, first introduced by Nedev and Muthukrishnan, can
be described as a game where two players -- Explorer and Director -- determine
the movement of a token on the vertices of a graph. At each time step, the
Explorer specifies a distance that the token must move hoping to maximize the
amount of vertices ultimately visited, and the Director adversarially chooses
where to move token in an effort to minimize this number. Given a graph and a
starting vertex, the number of vertices that are visited under optimal play is
denoted by $f_d(G,v)$.
  In this paper, we first reduce the study of $f_d (G,v)$ to the determination
of the minimal sets of vertices that are \textit{closed} in a certain
combinatorial sense, thus providing a structural understanding of each player's
optimal strategies. As an application, we address the problem on lattices and
trees. In the case of trees, we also provide a complete solution even in the
more restrictive setting where the strategy used by the Explorer is not allowed
to depend on their opponent's responses. In addition to this paper, a
supplementary companion note will be posted to arXiv providing additional
results about the game in a variety of specific graph families.
