On the Power of the Weisfeiler-Leman Test for Graph Motif Parameters

Seminal research in the field of graph neural networks (GNNs) has revealed a direct correspondence between the expressive capabilities of GNNs and the $k$-dimensional Weisfeiler-Leman ($k$WL) test, a widely-recognized method for verifying graph isomorphism. This connection has reignited interest in comprehending the specific graph properties effectively distinguishable by the $k$WL test.A central focus of research in this field revolves around determining the least dimensionality $k$, for which $k$WL can discern graphs with different number of occurrences of a pattern graph $p$. We refer to such a least $k$ as the WL-dimension of this pattern counting problem. This inquiry traditionally delves into two distinct counting problems related to patterns: subgraph counting and induced subgraph counting. Intriguingly, despite their initial appearance as separate challenges with seemingly divergent approaches, both of these problems are interconnected components of a more comprehensive problem: "graph motif parameters". In this paper, we provide a precise characterization of the WL-dimension of labeled graph motif parameters. As specific instances of this result, we obtain characterizations of the WL-dimension of the subgraph counting and induced subgraph counting problem for every labeled pattern $p$. Particularly noteworthy is our resolution of a problem left open in previous work concerning induced copies.We additionally demonstrate that in cases where the $k$WL test distinguishes between graphs with varying occurrences of a pattern $p$, the exact number of occurrences of $p$ can be computed uniformly using only local information of the last layer of a corresponding GNN.We finally delve into the challenge of recognizing the WL-dimension of various graph parameters. We give a polynomial time algorithm for determining the WL-dimension of the subgraph counting problem for given pattern $p$, answering an open question from previous work.We additionally show how to utilize deep results from the field of graph motif parameters, together with our characterization, to determine the WL-dimension of induced subgraph counting and counting $k$-graphlets.