[Paper Review] Oddomorphisms and homomorphism indistinguishability over graphs of bounded degree

David E. Roberson|arXiv (Cornell University)|Jun 21, 2022

Advanced Graph Theory Research4 citations

TL;DR

This paper introduces weak oddomorphisms—homomorphisms with parity constraints—and uses them to analyze homomorphism indistinguishability over families of bounded-degree graphs. It proves that homomorphism indistinguishability over bounded-degree graphs does not imply isomorphism, resolving a long-standing open question, and proposes conjectures linking minor-closure, union-closure, and distinct indistinguishability relations.

ABSTRACT

We introduce (weak) oddomorphisms of graphs which are homomorphisms with additional constraints based on parity. These maps turn out to have interesting properties (e.g., they preserve planarity), particularly in relation to homomorphism indistinguishability. Graphs $G$ and $H$ are *homomorphism indistinguishable* over a family $\mathcal{F}$ if $\hom(F,G) = \hom(F,H)$ for all $F \in \mathcal{F}$, where $\hom(F,G)$ is the number of homomorphisms from $F$ to $G$. A classical result of Lovász says that isomorphism is equivalent to homomorphism indistinguishability over the class of all graphs. In recent years it has been shown that many homomorphism indistinguishability relations have natural algebraic and/or logical formulations. Currently, much research in this area is focused on finding such reformulations. We aim to broaden the scope of current research on homomorphism indistinguishability by introducing new concepts/constructions and proposing several conjectures/questions. In particular, we conjecture that every family closed under disjoint unions and minors gives rise to a distinct homomorphism indistinguishability relation. We also show that if $\mathcal{F}$ is a family of graphs closed under disjoint unions, restrictions to connected components, and weak oddomorphisms, then $\mathcal{F}$ satisfies a certain maximality or closure property: homomorphism indistinguishability over $\mathcal{F}$ of $G$ and $H$ does not imply $\hom(F,G) = \hom(F,H)$ for any $F otin \mathcal{F}$. This allows us to answer a question raised over ten years ago, showing that homomorphism indistinguishability over graphs of bounded degree is not equivalent to isomorphism.

Motivation & Objective

• To expand the scope of homomorphism indistinguishability research beyond known characterizations by introducing new structural concepts.
• To investigate whether homomorphism indistinguishability over bounded-degree graphs implies isomorphism, a question open for over a decade.
• To propose and analyze new conjectures linking closure under minors, disjoint unions, and weak oddomorphisms to distinct indistinguishability relations.
• To establish a maximality property for families closed under weak oddomorphisms, showing that such families cannot be extended without altering indistinguishability.
• To explore the relationship between algebraic, logical, and topological properties of graph families and their corresponding homomorphism indistinguishability relations.

Proposed method

• Introduces weak oddomorphisms as homomorphisms satisfying parity constraints on preimage sizes of vertices.
• Uses the property that homomorphism counts factor over disjoint unions: $\hom(F_1 \cup F_2, G) = \hom(F_1, G) \cdot \hom(F_2, G)$.
• Applies the maximality result: if a family $\mathcal{F}$ is closed under disjoint unions, component restrictions, and weak oddomorphisms, then $\cong_{\mathcal{F}}$ does not imply $\hom(F,G) = \hom(F,H)$ for $F \notin \mathcal{F}$.
• Leverages Lovász’s classical result that isomorphism is equivalent to homomorphism indistinguishability over all graphs.
• Reduces the problem of distinguishing non-isomorphic graphs via bounded-degree families to the existence of graphs $H, H'$ with differing homomorphism counts only on graphs containing $K_n$ as a minor.
• Proposes that weak oddomorphisms may form a suborder of the minor order, suggesting a structural link between parity constraints and graph minors.

Experimental results

Research questions

• RQ1 Does homomorphism indistinguishability over graphs of bounded degree imply isomorphism?
• RQ2 Is every minor- and union-closed family of graphs associated with a distinct homomorphism indistinguishability relation?
• RQ3 Can the existence of weak oddomorphisms to complete graphs be used to characterize the presence of $K_n$ as a minor?
• RQ4 If a homomorphism indistinguishability relation is preserved under taking graph complements, is it necessarily equivalent to one defined by a minor-closed family?
• RQ5 What is the relationship between the complexity of deciding $\cong_{\mathcal{F}}$ and the structural properties of $\mathcal{F}$?

Key findings

• Homomorphism indistinguishability over graphs of bounded degree does not imply isomorphism, answering a question open for over ten years.
• The paper proves that if a family $\mathcal{F}$ is closed under disjoint unions, component restrictions, and weak oddomorphisms, then $\cong_{\mathcal{F}}$ is maximal: no $F \notin \mathcal{F}$ can be added without changing the indistinguishability relation.
• Conjecture 6 — that homomorphism counts differ only on graphs containing $K_n$ as a minor — is equivalent to Conjecture 5, which states that any proper minor- and union-closed family gives a non-isomorphism indistinguishability relation.
• The existence of weak oddomorphisms to $K_N$ may imply that a graph contains $K_n$ as a minor, suggesting a structural link between parity constraints and minor containment.
• The partial order of weak oddomorphisms may be a suborder of the reverse minor order, indicating a deeper algebraic-topological connection between homomorphisms and graph minors.
• The paper establishes that homomorphism indistinguishability over planar graphs is equivalent to quantum isomorphism, and over bounded treewidth graphs to the Sherali-Adams hierarchy, reinforcing the role of structural graph classes in indistinguishability.