🤖 AI Summary
This paper addresses the characterization of graph homomorphism indistinguishability over structurally restricted graph classes—such as graphs of bounded treewidth, bounded tree depth, and planar graphs—within a unified algebraic framework.
Method: It introduces, for the first time, a synthesis of homomorphism tensor embeddings with group representation theory and linear maps between tensor subspaces, integrating homomorphism counting, tensor algebra, and feasibility theory for linear equation systems.
Contribution/Results: (1) It provides exact algebraic characterizations of homomorphism indistinguishability over bounded-treewidth, bounded-pathwidth, and bounded-tree-depth graph classes; (2) it establishes its equivalence to the feasibility of specific linear systems (e.g., $Ax = b$); and (3) it unifies the algebraic foundations underlying logical equivalences (e.g., $C^k$, fixed-point logic with counting) and algebraic equivalences (e.g., Weisfeiler–Leman variants, matrix rank conditions), thereby resolving an open problem posed by Dell et al. concerning the characterization of bounded-pathwidth graphs.
📝 Abstract
Lov'asz (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 and graphs of bounded pathwidth, answering a question of Dell et al. (2018), and graphs of bounded treedepth.
