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On Endomorphism Universality of Sparse Graph Classes

Year: 2025
Type: article
Abstract: ABSTRACT We show that every commutative idempotent monoid (a.k.a. lattice) is the endomorphism monoid of a subcubic graph. This solves a problem of Babai and Pultr and the degree bound is best‐possible. On the other hand, we show that no class excluding a minor can have all commutative idempotent monoids among its endomorphism monoids. As a by‐product, we prove that monoids can be represented by graphs of bounded expansion (reproving a result of Nešetřil and Ossona de Mendez) and ‐cancellative monoids can be represented by graphs of bounded degree. Finally, we show that not all completely regular monoids can be represented by graphs excluding topological minor (strengthening a result of Babai and Pultr).
DOI: 10.1002/jgt.23262
Institutions: Centre National de la Recherche Scientifique, Université de Toulon, Aix-Marseille Université, Laboratoire d’Informatique et Systèmes, Universitat de Barcelona, Université Paris Sciences et Lettres, Lamsade, Université Paris Dauphine-PSL
Language: English
Open Access status: hybrid
Estimated APC paid: $4,150
Awards: G195474297, G3073974760, Metric graph theory, DAGs and Digraph Decompositions, G4274939422, G6468150428, G6803667113, G6989706184
OpenAlex ID: W4411214464
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Cites: 41
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Related to: 10