Logical complexity of graphs: a survey

TL;DR

This survey explores the logical complexity parameters of finite graphs, such as definability, depth, width, and length, and their connections to graph isomorphism, random graph evolution, and logical restrictions.

Contribution

It compiles and discusses known estimates of graph logical parameters and examines their behavior under various logical restrictions and extensions.

Findings

Logical depth, width, and length relate to graph isomorphism testing.

Behavior of graph definability changes with logical restrictions.

Connections between logical complexity and probabilistic graph properties.

Abstract

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).