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Logical Equivalences, Homomorphism Indistinguishability, and Forbidden Minors

Tim Seppelt

TL;DR

This work develops a unifying framework linking homomorphism indistinguishability and logic on graphs, proving that for any self-complementary logic with a hom-indistinguishability characterisation there exists a minor-closed graph class capturing the same equivalence relations. It establishes a general correspondence: closure properties of a graph class translate into preservation properties of its indistinguishability relation, enabling the transfer of graph-minor theory results to the expressive power of logics. A central result classifies all essentially profinite HD-closed graph classes, yielding broad implications for Roberson-style questions and the limits of certain graph equivalences. The paper also analyzes the complexity of HomInd over these classes and derives cancellation laws, linking algebraic structure to logical distinguishability and quantum isomorphism phenomena, with practical impact on graph kernels, motif counting, and database-style bag semantics.

Abstract

Two graphs $G$ and $H$ are homomorphism indistinguishable over a class of graphs $\mathcal{F}$ if for all graphs $F \in \mathcal{F}$ the number of homomorphisms from $F$ to $G$ is equal to the number of homomorphisms from $F$ to $H$. Many natural equivalence relations comparing graphs such as (quantum) isomorphism, spectral, and logical equivalences can be characterised as homomorphism indistinguishability relations over certain graph classes. Abstracting from the wealth of such instances, we show in this paper that equivalences w.r.t. any self-complementarity logic admitting a characterisation as homomorphism indistinguishability relation can be characterised by homomorphism indistinguishability over a minor-closed graph class. Self-complementarity is a mild property satisfied by most well-studied logics. This result follows from a correspondence between closure properties of a graph class and preservation properties of its homomorphism indistinguishability relation. Furthermore, we classify all graph classes which are in a sense finite (essentially profinite) and satisfy the maximality condition of being homomorphism distinguishing closed, i.e. adding any graph to the class strictly refines its homomorphism indistinguishability relation. Thereby, we answer various questions raised by Roberson (2022) on general properties of the homomorphism distinguishing closure.

Logical Equivalences, Homomorphism Indistinguishability, and Forbidden Minors

TL;DR

This work develops a unifying framework linking homomorphism indistinguishability and logic on graphs, proving that for any self-complementary logic with a hom-indistinguishability characterisation there exists a minor-closed graph class capturing the same equivalence relations. It establishes a general correspondence: closure properties of a graph class translate into preservation properties of its indistinguishability relation, enabling the transfer of graph-minor theory results to the expressive power of logics. A central result classifies all essentially profinite HD-closed graph classes, yielding broad implications for Roberson-style questions and the limits of certain graph equivalences. The paper also analyzes the complexity of HomInd over these classes and derives cancellation laws, linking algebraic structure to logical distinguishability and quantum isomorphism phenomena, with practical impact on graph kernels, motif counting, and database-style bag semantics.

Abstract

Two graphs and are homomorphism indistinguishable over a class of graphs if for all graphs the number of homomorphisms from to is equal to the number of homomorphisms from to . Many natural equivalence relations comparing graphs such as (quantum) isomorphism, spectral, and logical equivalences can be characterised as homomorphism indistinguishability relations over certain graph classes. Abstracting from the wealth of such instances, we show in this paper that equivalences w.r.t. any self-complementarity logic admitting a characterisation as homomorphism indistinguishability relation can be characterised by homomorphism indistinguishability over a minor-closed graph class. Self-complementarity is a mild property satisfied by most well-studied logics. This result follows from a correspondence between closure properties of a graph class and preservation properties of its homomorphism indistinguishability relation. Furthermore, we classify all graph classes which are in a sense finite (essentially profinite) and satisfy the maximality condition of being homomorphism distinguishing closed, i.e. adding any graph to the class strictly refines its homomorphism indistinguishability relation. Thereby, we answer various questions raised by Roberson (2022) on general properties of the homomorphism distinguishing closure.
Paper Structure (16 sections, 35 theorems, 29 equations, 1 figure, 1 table)

This paper contains 16 sections, 35 theorems, 29 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Closure Properties Correspond to Preservation Properties
  4. Taking Summands and Preservation under Disjoint Unions
  5. Taking Minors and Preservation under Complements
  6. Taking Subgraphs and Preservation under Full Complements
  7. Taking Induced Subgraphs, Contracting Edges, and Lexicographic Products
  8. Applications
  9. Equivalences over Self-Complementary Logics
  10. Examples for Self-Complementary Logics
  11. Applications: Graph Minor Theory and Expressive Power
  12. Classification of Homomorphism Distinguishing Closed Essentially Profinite Graph Classes
  13. Applications of \ref{['thm:profinite']}
  14. Complexity of Homomorphism Indistinguishability over Essentially Profinite Graph Classes
  15. Cancellation Laws
  16. ...and 1 more sections

Key Result

Theorem 1

Let $(\mathsf{L}, \models)$ be a self-complementary logic on graphs for which there exists a graph class $\mathcal{F}$ such that two graphs $G$ and $H$ are homomorphism indistinguishable over $\mathcal{F}$ if and only if they are $\mathsf{L}$-equivalent. Then there exists a minor-closed graph class

Figures (1)

  • Figure 1: Relationship between closure properties of homomorphism distinguishing closed graph classes. The non-obvious implications are dashed and proven in \ref{['lem:minors', 'lem:contr']}.

Theorems & Definitions (85)

  • Theorem 1: restate=thmLogic,name=
  • Theorem 2
  • Theorem 3
  • Lemma 4
  • proof
  • Lemma 6
  • proof
  • Theorem 7
  • proof
  • Theorem 8
  • ...and 75 more