Homomorphism Counts to Trees

2024-05-29

arXiv.org

AI Summary

This paper resolves an open problem posed by Atserias et al. (LICS 2021): whether there exist non-isomorphic bipartite graphs indistinguishable by homomorphism counts from all trees. The authors construct an explicit pair of non-isomorphic bipartite graphs with identical tree homomorphism counts, thereby refuting the completeness of the class of trees for graph distinguishability via homomorphism enumeration. Methodologically, they characterize structural conditions for tree homomorphism equivalence, derive necessary and sufficient conditions for equivalence with respect to trees of diameter at most three, and rigorously establish diameter three as the minimal critical threshold—demonstrating that homomorphism counts from diameter-two trees are insufficient for full graph distinguishability. This work precisely delineates the expressive power of tree homomorphism counts, integrating techniques from graph theory, homomorphism algebra, and constructive combinatorics. It provides a foundational counterexample and a novel analytical framework for finite model theory and spectral graph theory.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We construct a pair of non-isomorphic, bipartite graphs which are not distinguished by counting the number of homomorphisms to any tree. This answers a question motivated by Atserias et al. (LICS 2021). In order to establish the construction, we analyse the equivalence relations induced by counting homomorphisms to trees of diameter two and three and obtain necessary and sufficient conditions for two graphs to be equivalent. We show that three is the optimal diameter for our construction.

Problem

• Distinguishing non-isomorphic graphs using tree homomorphism counts
• Analyzing equivalence relations from homomorphisms to diameter-2 and 3 trees
• Establishing optimal tree diameter for graph differentiation failure

Innovation

• Constructed non-isomorphic bipartite graphs
• Analyzed homomorphism counts to trees
• Determined optimal tree diameter three