Bounding the Weisfeiler-Leman Dimension via a Depth Analysis of I/R-Trees

TL;DR

This work resolves long-standing questions about the descriptive complexity of graphs under counting logic by establishing a near-quadratic improvement in the WL dimension for general $n$-vertex graphs: any graph on $n$ vertices is definable in ${\sf C}^{k}$ with $k = \frac{n}{4} + o(n)$ variables. The authors introduce the WL depth, a novel framework that extends the Individualization/ Refinement paradigm with component-splitting and edge-flip operations, linking depth to WL dimension via a general bound. They obtain a vertex-cover–dependent bound ${\mathrm{WL\text{-}dim}}(G) \le \frac{2}{3}r + 3$ and, more ambitiously, prove a $2$-WL depth bound of $\frac{n}{4} + o(n)$, together yielding the main upper bound. A complementary lower bound via the CFI construction shows WL dimension can be as large as $\frac{1}{96}n - o(n)$, highlighting a remaining gap. Overall, the WL depth approach offers a promising route to tighter WL bounds and deepens the connection between isomorphism testing techniques and logical definability.

Abstract

The Weisfeiler-Leman (WL) dimension is an established measure for the inherent descriptive complexity of graphs and relational structures. It corresponds to the number of variables that are needed and sufficient to define the object of interest in a counting version of first-order logic (FO). These bounded-variable counting logics were even candidates to capture graph isomorphism, until a celebrated construction due to Cai, Fürer, and Immerman [Combinatorica 1992] showed that $Ω(n)$ variables are required to distinguish all non-isomorphic $n$-vertex graphs. Still, very little is known about the precise number of variables required and sufficient to define every $n$-vertex graph. For the bounded-variable (non-counting) FO fragments, Pikhurko, Veith, and Verbitsky [Discret. Appl. Math. 2006] provided an upper bound of $\frac{n+3}{2}$ and showed that it is essentially tight. Our main result yields that, in the presence of counting quantifiers, $\frac{n}{4} + o(n)$ variables suffice. This shows that counting does allow us to save variables when defining graphs. As an application of our techniques, we also show new bounds in terms of the vertex cover number of the graph. To obtain the results, we introduce a new concept called the WL depth of a graph. We use it to analyze branching trees within the Individualization/Refinement (I/R) paradigm from the domain of isomorphism algorithms. We extend the recursive procedure from the I/R paradigm by the possibility of splitting the graphs into independent parts. Then we bound the depth of the obtained branching trees, which translates into bounds on the WL dimension and thereby on the number of variables that suffice to define the graphs.

Figures (2)

• Figure 1: A $1$-IRC tree of $C_6$. Every gray region corresponds to one node $t$ of the tree with the colored graph $\gamma(t)$ drawn inside. The root node is the only individualization node. Hence, the individualization depth of the $1$-IRC tree is equal to $1$.
• Figure 2: A colored graph $(G,\chi)$ and a vertex cover $S$ of $G$. We set $I \coloneqq V \setminus S$. The set $\chi(S)$ contains the colors red, blue, green and pink. The set $\chi(S) \setminus \chi(I)$ contains only green and pink. Hence, $p = 4 + 2 = 6$ in this example.

Theorems & Definitions (87)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 2.1: CaiFI92ImmermanL90
• Lemma 2.2
• proof
• Definition 3.1
• Example 3.2
• Example 3.3
• Example 3.4