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We define a family of vertex colouring games played over a pair of graphs or digraphs    (   G  ,  H   )    by players   ∀   and   ∃  . These games arise from work on a longstanding open problem in algebraic logic. It is conjectured that there is a natural number   n   such that   ∀   always has a winning strategy in the game with   n   colours whenever   G  ≇  H  . This is related to the reconstruction conjecture for graphs and the degree‐associated reconstruction conjecture for digraphs. We show that the reconstruction conjecture implies our game conjecture with   n  =  3   for graphs, and the same is true for the degree‐associated reconstruction conjecture and our conjecture for digraphs. We show (for any   k  <  ω  ) that the 2‐colour game can distinguish certain nonisomorphic pairs of graphs that cannot be distinguished by the   k  ‐dimensional Weisfeiler–Leman algorithm. We also show that the 2‐colour game can distinguish the nonisomorphic pairs of graphs in the families defined by Stockmeyer as counterexamples to the original digraph reconstruction conjecture.

Seurat games on Stockmeyer graphs