Homomorphism Tensors and Linear Equations

DROPS (Schloss Dagstuhl – Leibniz Center for Informatics)

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 and representation-theoretic 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.

Authors 3

1. Martin Grohe Aachen

   RWTH Aachen University

   Affiliation as printed

   RWTH Aachen University, Germany

2. Gaurav Rattan Aachen

   RWTH Aachen University

   Affiliation as printed

   RWTH Aachen University, Germany

3. Tim Seppelt Aachen

   RWTH Aachen University

   Affiliation as printed

   RWTH Aachen University, Germany