The Explorer–Director Game on Graphs

The Explorer-Director game, first introduced by Nedev and Muthukrishnan, can be described as a game where two players—Explorer Director—determine the movement of token that is positioned on vertices some given graph. At each time step, Explorer specifies distance must move with an aim to maximize total number ultimately visited. However, Director adversarially chooses in effort minimize this number. ends when no new Given graph G starting vertex v, are visited under optimal play denoted $$f_d(G,v)$$ . In paper, we reduce study $$f_d (G,v)$$ determination minimum sets closed certain combinatorial sense, thus providing structural understanding player’s strategies. As application, provide exact results well more general bounds square lattice or tree. case trees, also complete solution even restrictive setting strategy used not allowed depend their opponent’s responses. addition supplementary companion note will posted arXiv additional about variety specific families.