A Dense Weisfeiler-Leman Algorithm for Deciding Bounded-Cliquewidth Homomorphism Indistinguishability

2026-08-13

AI Summary

This study addresses the decidability of homomorphism indistinguishability over classes of graphs with bounded clique-width—a question previously unresolved for dense graph classes. To this end, the authors introduce the dense Weisfeiler–Leman algorithm, which colors k-tuples of vertex subsets rather than k-tuples of vertices, thereby effectively capturing the homomorphism profiles of graphs with clique-width at most k. The approach is further extended to CMSO₁-definable classes of bounded clique-width. The main contributions include the first proof that homomorphism indistinguishability is decidable for bounded clique-width graph classes; additionally, for bounded linear clique-width classes, the problem lies in PSPACE and admits PSPACE-complete instances, thus overcoming prior limitations that applied only to sparse graph classes.

Abstract

Two graphs $G$ and $H$ are homomorphism indistinguishable over a graph class $\mathcal{F}$ if they admit the same number of homomorphisms from every graph in $\mathcal{F}$. A wide range of relaxations of graph isomorphism arise this way: isomorphism itself over the class of all graphs [Lovász, Acta Math. Hung. 1967], equivalence under the $k$-dimensional Weisfeiler-Leman algorithm over the graphs of treewidth $\leq k$ [Dvořák, J. Graph Theory 2010], and quantum isomorphism over planar graphs [Mančinska-Roberson, FOCS 2020]. Since the class $\mathcal{F}$ is typically infinite, it is not clear a priori whether homomorphism indistinguishability over $\mathcal{F}$ is decidable; for planar graphs it is undecidable. Every class for which decidability was previously known is sparse. We give the first decidability results for dense graph classes: We introduce the dense Weisfeiler-Leman algorithm that decides homomorphism indistinguishability over the class of graphs of cliquewidth $\leq k$, the dense counterpart of treewidth. This relation was not previously known to be decidable. The algorithm colors $k$-tuples of vertex subsets rather than $k$-tuples of vertices. Beyond the class of all graphs of cliquewidth $\leq k$, we prove a general meta-theorem: homomorphism indistinguishability over every $\mathsf{CMSO}_1$-definable graph class of bounded cliquewidth is decidable, in randomized exponential time. For classes of bounded linear cliquewidth the bound improves to $\mathsf{PSPACE}$, and we show this is tight by exhibiting such a class for which the problem is $\mathsf{PSPACE}$-complete. These are the first general algorithms for homomorphism indistinguishability over dense graph classes.

Problem

• homomorphism indistinguishability
• cliquewidth
• decidability
• dense graph classes
• graph isomorphism

Innovation

• dense Weisfeiler-Leman
• homomorphism indistinguishability
• bounded cliquewidth
• CMSO_1-definable
• PSPACE-completeness