On Testability of First-Order Properties in Bounded-Degree Graphs

TL;DR

This paper investigates which first-order properties of bounded-degree graphs can be tested efficiently, showing that properties with certain logical structures are testable with constant queries, while others are not, extending known results from dense graphs.

Contribution

It characterizes the testability of first-order properties in bounded-degree graphs based on their logical quantifier structure and introduces new lower bounds using expander graph constructions.

Findings

FO properties with $orall^* oorall^*$ are testable with constant queries

FO properties with $orall^* oorall^*$ are not testable in general

Certain neighborhood-based properties are testable under mild degree assumptions

Abstract

We study property testing of properties that are definable in first-order logic (FO) in the bounded-degree graph and relational structure models. We show that any FO property that is defined by a formula with quantifier prefix $\exists^*\forall^*$ is testable (i.e., testable with constant query complexity), while there exists an FO property that is expressible by a formula with quantifier prefix $\forall^*\exists^*$ that is not testable. In the dense graph model, a similar picture is long known (Alon, Fischer, Krivelevich, Szegedy, Combinatorica 2000), despite the very different nature of the two models. In particular, we obtain our lower bound by a first-order formula that defines a class of bounded-degree expanders, based on zig-zag products of graphs. We expect this to be of independent interest. We then prove testability of some first-order properties that speak about isomorphism types of neighbourhoods, including testability of $1$-neighbourhood-freeness, and $r$-neighbourhood-freeness under a mild assumption on the degrees.