  We define a general framework of partition games for formulating two-player
pebble games over finite structures. We show that one particular such game,
which we call the invertible-map game, yields a family of polynomial-time
approximations of graph isomorphism that is strictly stronger than the
well-known Weisfeiler-Lehman method. The general framework we introduce
includes as special cases the pebble games for finite-variable logics with and
without counting. It also includes a matrix-equivalence game, introduced here,
which characterises equivalence in the finite-variable fragments of matrix-rank
logic. We show that the equivalence defined by the invertible-map game is a
refinement of the equivalence defined by each of these games for
finite-variable logics.
