Homomorphism Tensors and Linear Equations

Martin Grohe, Gaurav Rattan, Tim Seppelt

Research output: Journal Article or Conference Article in JournalJournal articleResearchpeer-review

Abstract

Lovász (1967) showed that two graphs 𝐺 and 𝐻 are isomorphic if and only if
they are homomorphism indistinguishable over the class of all graphs, i.e. for every graph 𝐹, the number of homomorphisms from 𝐹 to 𝐺 equals the number of homomorphisms from 𝐹 to 𝐻. Recently, homomorphism indistinguishability over restricted classes of graphs such as bounded treewidth, bounded treedepth and planar graphs, has emerged as a surprisingly powerful framework for capturing diverse equivalence relations on graphs arising from logical equivalence and algebraic equation systems.

In this paper, we provide a unified algebraic framework for such results by examining the linear-algebraic structure of tensors counting homomorphisms from labelled graphs. The existence of certain linear transformations between such homomorphism tensor subspaces can be interpreted both as homomorphism indistinguishability over a graph class and as feasibility of an equational system. Following this framework, we obtain characterisations of homomorphism indistinguishability over several natural graph classes, namely trees of bounded degree, graphs of bounded pathwidth (answering a question of Dell et al. (2018)), and graphs of bounded treedepth.
Original languageEnglish
JournalAdvances in Combinatorics
ISSN2517-5599
DOIs
Publication statusPublished - 2 Apr 2025

Keywords

  • Graph homomorphisms
  • Homomorphism indistinguishability
  • Labelled graphs
  • Treewidth
  • Pathwidth
  • Linear equations
  • Sherali–Adams relaxation
  • Specht–Wiegmann Theorem
  • Weisfeiler–Leman algorithm

Fingerprint

Dive into the research topics of 'Homomorphism Tensors and Linear Equations'. Together they form a unique fingerprint.

Cite this