by $k$-WL. The game is an Ehrenfeucht-Fra[ï]{}ss[é]{}-type game with two players Duplicator and Spoiler. Initially $k+1$ pairs of pebbles, each pair uniquely colored, are placed next to two given input graphs $\Gamma_1,\Gamma_2$. Each round proceeds as follows: Spoiler picks up a pebble pair $(p_i; p_i')$ of pebbles of the same color. Then Duplicator chooses a bijection $\varphi$ from $V(\Gamma_1)$ to $V(\Gamma_2)$. Then Spoiler places pebble $p_i$ on a vertex $v\in V(\Gamma_1)$ and places $p_i'$ on $\varphi(v)$. Spoiler wins if at any point in time the graph induced by the vertices occupied by pebbles in $V(\Gamma_1)$ is not isomorphic to the graph induced by the vertices occupied by pebbles in $V(\Gamma_2)$ via a map that sends a pebble $p_i$ to its corresponding pebble of the same color $p_i'$ in the other graph. Spoiler also wins (in round 0) if $|V(\Gamma_1)|\neq |V(\Gamma_2)|$.

When using $k+1$ pebbles on two graphs, the game can be won by Spoiler exactly if $k$-WL distinguishes the graphs [@DBLP:journals/iandc/Hella96].

WL-type algorithms on groups
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As with graphs we would like to be able to study combinatorial properties of finite groups using WL-type algorithms. The natural approach is to adapt the methods from the last section to suit (finite) 