Graphs Identified by Logics with Counting

TL;DR

This paper classifies which graphs and structures are uniquely identified by the two-variable counting logic C2, enabling efficient decision procedures and revealing properties of automorphisms and partitions within these structures.

Contribution

It provides a classification of structures identified by C2, an almost linear time decision algorithm, and insights into automorphisms and partitions for these structures.

Findings

Structures identified by C2 have specific automorphism properties.

Deciding if a structure is identified by C2 can be done in almost linear time.

For k>2, some properties do not hold, as shown by counterexamples.

Abstract

We classify graphs and, more generally, finite relational structures that are identified by C2, that is, two-variable first-order logic with counting. Using this classification, we show that it can be decided in almost linear time whether a structure is identified by C2. Our classification implies that for every graph identified by this logic, all vertex-colored versions of it are also identified. A similar statement is true for finite relational structures. We provide constructions that solve the inversion problem for finite structures in linear time. This problem has previously been shown to be polynomial time solvable by Martin Otto. For graphs, we conclude that every C2-equivalence class contains a graph whose orbits are exactly the classes of the C2-partition of its vertex set and which has a single automorphism witnessing this fact. For general k, we show that such statements are not true by providing examples of graphs of size linear in k which are identified by C3 but for which the orbit partition is strictly finer than the Ck-partition. We also provide identified graphs which have vertex-colored versions that are not identified by Ck.