Furthermore we write $WL_k^J\preceq WL_{k'}^{J'}$ if it holds that $G\equiv_{WL_{k'}^{J'}} H\Rightarrow G\equiv_{WL_k^J} H$, i.e., the distinguishing power of $WL_k^J$ is weaker than or equal to the distinguishing power of $WL_{k'}^{J'}$.

The main result of this section is that we can exchange one version for another when we multiply the dimension with a constant factor. When studying WL-type algorithms it is often useful to have equivalent pebble games at hand, so we first associate a pebble game to each of the variants above.

Bijective $k$-pebble games
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We now define suitable pebble games for the different versions. Each of these games is played by two players Spoiler and Duplicator and in each case we will say that Duplicator wins the game if and only if there is a strategy for Duplicator to keep the game going on forever. The board consists of a pair of finite groups $G,H$ of equal order (or rather their elements) or a pair of corresponding graphs $\Gamma_G,\Gamma_H$ for Version III, respectively. There are $k$ pairs of pebbles $(p_1;p'_1), \ldots (p_k;p'_k)$. We think of pebbles in the same pair as having the same color, and pebbles from different pairs as having distinct colors. The pebbles can be 