Logical complexity of graphs: a survey

We discuss the definability of finite graphs in first-order logic with two relation symbols for adjacency and equality of vertices. The logical depth $D(G)$ of a graph $G$ is equal to the minimum quantifier depth of a sentence defining $G$ up to isomorphism. The logical width $W(G)$ is the minimum number of variables occurring in such a sentence. The logical length $L(G)$ is the length of a shortest defining sentence. We survey known estimates for these graph parameters and discuss their relations to other topics (such as the efficiency of the Weisfeiler-Lehman algorithm in isomorphism testing, the evolution of a random graph, quantitative characteristics of the zero-one law, or the contribution of Frank Ramsey to the research on Hilbert’s Entscheidungsproblem). Also, we trace the behavior of the descriptive complexity of a graph as the logic becomes more restrictive (for example, only definitions with a bounded number of variables or quantifier alternations are allowed) or more expressible (after powering with counting quantifiers).

💡 Research Summary

The paper surveys the descriptive complexity of finite graphs when they are defined in first‑order logic (FO) that uses only two binary relation symbols: adjacency and equality of vertices. Three quantitative measures are central: logical depth $D(G)$, the minimum quantifier‑nesting depth of a sentence that uniquely defines $G$ up to isomorphism; logical width $W(G)$, the smallest number of variables required; and logical length $L(G)$, the length of the shortest defining sentence. The authors collect known upper and lower bounds for these parameters, examine how they behave for special families of graphs, and relate them to several major topics in theoretical computer science and combinatorics.

First, the paper formalises the connection between $D(G)$ and the Ehrenfeucht–Fraïssé (EF) game: $D(G)$ equals the minimum number of rounds in which Spoiler can force a win against Duplicator on $G$ versus any non‑isomorphic graph. Similarly, $W(G)$ corresponds to the smallest $k$ such that FO$^{k}$ (the $k$‑variable fragment) can separate $G$ from all non‑isomorphic graphs. $L(G)$ is treated as a compression measure, reflecting how succinctly a graph can be described in FO. Trivial bounds $D(G)\le n+1$, $W(G)\le n$, $L(G)\le 2^{O(n)}$ hold for any $n$‑vertex graph, but the survey emphasizes much tighter results for concrete classes. Complete graphs, complete bipartite graphs, and cycles have $D(G)=2$ or $\Theta(\log n)$, while trees and bounded‑degree graphs often admit $D(G)=O(\log n)$.

A substantial portion of the survey is devoted to the Weisfeiler–Lehman (WL) algorithm. The $k$‑dimensional WL refinement is shown to be equivalent in power to FO$^{k}$; consequently, if $W(G)\le k$, the $k$‑WL procedure distinguishes $G$ from any non‑isomorphic graph. The authors discuss empirical findings that 2‑WL (color refinement) and 3‑WL already separate the vast majority of real‑world graph instances, explaining why $W(G)$ is typically very small in practice.

The behavior of $D(G)$, $W(G)$, and $L(G)$ in the Erdős–Rényi random graph model $G(n,p)$ is analysed in detail. For constant $p$, almost all graphs satisfy $D(G)=\Theta(\log n)$ and $W(G)=\Theta(\log n)$, while $L(G)=\Theta(n\log n)$. When $p$ falls below $n^{-1}$, the random graph becomes tree‑like and the depth drops to $O(\log\log n)$. These probabilistic results are linked to quantitative versions of the zero‑one law: in fragments of FO with bounded quantifier alternation, the probability that a random graph satisfies a given sentence converges rapidly, and the convergence rate is governed by the typical values of $D(G)$ and $W(G)$.

The historical interlude on Frank Ramsey highlights his early work on the Entscheidungsproblem and graph colourings, which implicitly produced families of graphs with large logical depth. The survey explains how Ramsey’s constructions give lower bounds for $D(G)$ and $W(G)$, illustrating the deep interplay between combinatorial extremal results and logical definability.

The authors then explore how restricting the logic changes the descriptive complexity. Bounding the number of variables or the number of quantifier alternations forces $L(G)$ to grow exponentially for many graphs, while allowing counting quantifiers (∃≥k) or modular quantifiers dramatically reduces both $D(G)$ and $W(G)$. In particular, the two‑variable counting logic $C^{2}$ can define almost every graph with constant depth and width, mirroring the power of the 2‑WL algorithm enhanced with counting.

Finally, the paper lists open problems: determining tighter average‑case bounds for $D(G)$ and $W(G)$, characterising graphs that achieve the worst‑case depth, understanding the precise relationship between higher‑dimensional WL and extensions of FO with counting or modular operators, and extending the analysis to dynamic or massive networks. The survey thus provides a comprehensive map of the current knowledge on graph logical complexity and points to promising directions for future research.