  We define a family of vertex colouring games played over a pair of graphs or
digraphs $(G,H)$ by players $\forall$ and $\exists$. 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 $\forall$ always has a winning strategy
in the game with $n$ colours whenever $G\not\cong H$. 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 $k<\omega$) that the 2-colour game can distinguish certain
non-isomorphic 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 non-isomorphic pairs of graphs in the families defined by
Stockmeyer as counterexamples to the original digraph reconstruction
conjecture.
